The expression ‘2x2 + 8x + x2 = 16′ represents a quadratic equation. The task involves finding the value of ‘x’ that satisfies this equation. Simplifying the equation leads to a standard quadratic form which can then be solved using various methods such as factoring, completing the square, or applying the quadratic formula. In this instance, combining like terms results in 3x2 + 8x = 16 or 3x2 + 8x – 16 = 0.
Solving quadratic equations is a fundamental skill in algebra and has applications in various fields, including physics, engineering, and economics. Understanding how to determine the roots of a quadratic equation is essential for modeling and solving real-world problems. The process of finding the root provides insights into the behavior of the quadratic function and the nature of its solutions. Historically, quadratic equations have been studied for centuries, with methods for their solution dating back to ancient civilizations.
The subsequent sections will delve into the process of obtaining the specific solution, providing a step-by-step explanation of the most efficient method for solving the quadratic equation 3x2 + 8x – 16 = 0 and verifying that solution. This involves a careful examination of the equation’s coefficients and the application of appropriate algebraic techniques to arrive at the definitive answer.
1. Quadratic Equation
The phrase “what is the only solution of 2x2 + 8x + x2 = 16″ directly pertains to the realm of quadratic equations. A quadratic equation is a polynomial equation of the second degree. The fundamental aim is to identify the values of the variable, in this case ‘x’, that, when substituted into the equation, make the equation true. The existence of a single solution, as implied by the phrase, dictates the nature of the quadratic equation’s roots, specifically indicating a repeated real root. This implies that the discriminant of the quadratic equation, when arranged in the standard form ax2 + bx + c = 0, must be equal to zero (b2 – 4ac = 0). The correct classification as a quadratic equation is therefore a prerequisite for implementing solution methodologies and interpreting the result.
Consider the broader context: quadratic equations model diverse phenomena, from projectile motion in physics to optimization problems in economics. The ability to determine the solution of a given quadratic equation has significant implications. For instance, if the equation modeled the trajectory of a projectile, the solution might represent the point at which the projectile impacts the ground. If there were only one valid solution, this would translate to a specific condition of launch, angle, and target placement. In engineering, quadratic equations can describe the bending of beams, and their solution informs the safety and structural integrity of designs. Hence, accurately identifying a given problem as a quadratic equation provides the foundation for effective problem-solving across numerous disciplines.
In summary, recognizing the problem “what is the only solution of 2x2 + 8x + x2 = 16″ as a quadratic equation is paramount. It dictates the appropriate mathematical tools and techniques to be employed, ultimately shaping the interpretation and application of the resulting solution. The presence of a “only solution” further clarifies the specific type of quadratic equation and its mathematical properties, emphasizing the importance of precise methodology to find and verify that solution.
2. Coefficient Identification
Coefficient identification is a critical initial step in determining the solution of quadratic equations, especially when tasked with finding “what is the only solution of 2x2 + 8x + x2 = 16″. Accurate identification ensures the subsequent application of appropriate solution methods and contributes directly to obtaining a correct and verifiable answer.
-
Standard Form Transformation
Before coefficients can be identified, the equation must be transformed into its standard quadratic form, ax2 + bx + c = 0. In this case, the original expression, 2x2 + 8x + x2 = 16, must be simplified and rearranged to 3x2 + 8x – 16 = 0. The coefficients are then readily apparent: a = 3, b = 8, and c = -16. Failure to correctly transform the equation leads to misidentification of coefficients, inevitably causing errors in the solution process.
-
Impact on Factoring
When factoring is the chosen method of solution, the coefficients directly influence the factoring process. Specifically, the ‘a’ and ‘c’ coefficients determine the possible combinations of factors that need to be considered. For the equation 3x2 + 8x – 16 = 0, the factors of 3 and -16 dictate potential binomial expressions. Incorrect coefficient identification results in the exploration of inappropriate factor combinations, preventing successful factorization and subsequent root determination.
-
Quadratic Formula Dependence
The quadratic formula, x = (-b (b2 – 4ac)) / (2a), explicitly utilizes the identified coefficients. Substituting incorrect values for ‘a’, ‘b’, or ‘c’ will result in a flawed calculation, leading to an incorrect solution. Given that the problem statement stipulates “the only solution,” any error in coefficient identification directly contradicts this condition, highlighting the importance of accuracy in this foundational step.
-
Discriminant Influence
The discriminant, b2 – 4ac, a component of the quadratic formula, determines the nature of the roots. Specifically, the discriminant of zero indicates a single, repeated real root, aligning with the problem’s emphasis on a unique solution. The correct identification of ‘a’, ‘b’, and ‘c’ is essential for calculating the discriminant accurately. An incorrect discriminant value, stemming from coefficient misidentification, may lead to the conclusion that the equation has two distinct real roots or no real roots, thereby conflicting with the problem’s stated condition.
In conclusion, coefficient identification constitutes a fundamental prerequisite for solving quadratic equations, particularly when seeking the unique solution to 2x2 + 8x + x2 = 16. Its influence permeates all solution methodologies, from factoring to the application of the quadratic formula, and directly impacts the accuracy and validity of the derived root. Therefore, meticulous and accurate coefficient identification is paramount to successfully addressing the problem statement and obtaining the correct, single solution.
3. Root Determination
Root determination is the central objective when presented with “what is the only solution of 2x2 + 8x + x2 = 16″. The phrase necessitates identifying the specific numerical value, or root, of ‘x’ that satisfies the given equation. The existence of “only” one solution significantly constrains the possible outcomes and dictates the methodologies employed for its identification. Successful root determination validates the equation’s equality; substituting the derived value for ‘x’ should result in both sides of the equation being equal. The implication of a single solution directly impacts the selection and implementation of solution strategies, demanding precision and accuracy in the application of algebraic techniques. For example, if a real-world problem modeled by this equation concerned the dimensions of a square where ‘x’ represented the length of a side, the root would provide the precise measurement fulfilling the specified area condition.
The process of root determination relies heavily on transforming the original equation into a solvable form, often the standard quadratic form (ax2 + bx + c = 0). Factoring, completing the square, or utilizing the quadratic formula are all potential methods for root determination. The choice of method is influenced by the specific characteristics of the equation and the solver’s preference. Each method aims to isolate ‘x’ and determine its numerical value. Given that “only” one solution is expected, verification through substitution back into the original equation becomes paramount. This process serves as a crucial validation step, confirming the accuracy of the derived root and ensuring it aligns with the initial equation’s constraints. If the equation represented a critical parameter in a machine’s operation, a correctly determined root would guarantee the machine functions as designed; an incorrect root could lead to malfunction or failure.
In conclusion, root determination is intrinsically linked to the core of “what is the only solution of 2x2 + 8x + x2 = 16″. It represents the culmination of algebraic manipulation and problem-solving skills aimed at isolating the unique value of ‘x’. The specification of a single solution mandates a focus on accurate methodologies and thorough verification processes. Challenges in root determination often arise from algebraic errors or the misapplication of solution techniques. Understanding the interconnectedness between the problem statement and the fundamental goal of root determination is essential for achieving a correct and validated outcome, with implications spanning various real-world applications where quadratic equations provide valuable models and insights.
4. Factoring Methods
Factoring methods represent a specific class of techniques applicable to solving quadratic equations, offering a direct route to finding the solutions, particularly when addressing the challenge posed by “what is the only solution of 2x2 + 8x + x2 = 16″. Their efficacy hinges on the equation’s structure and whether it can be readily decomposed into a product of simpler expressions.
-
Factorability and Equation Structure
The success of factoring depends on whether the quadratic expression can be factored into two binomial expressions with integer or rational coefficients. In the context of 2x2 + 8x + x2 = 16, which simplifies to 3x2 + 8x – 16 = 0, the equation may or may not be easily factorable. If the discriminant (b2 – 4ac) is a perfect square, factoring is generally feasible. However, if the discriminant is not a perfect square, alternative methods like the quadratic formula become more suitable. The unique solution constraint influences the factoring outcome, implying the factored form will likely lead to a repeated factor.
-
Factoring by Decomposition
The decomposition method, when applicable, involves rewriting the middle term of the quadratic equation as a sum or difference of two terms, allowing for subsequent factoring by grouping. For 3x2 + 8x – 16 = 0, the middle term ‘8x’ needs to be decomposed into two terms whose coefficients multiply to (3 * -16 = -48) and add up to 8. This decomposition, if possible, directly leads to the binomial factors. Finding this decomposition is key to directly obtaining the roots. In scenarios where a unique solution is desired, the resulting factors should, after simplification, be identical or multiples of each other, indicating a repeated root.
-
Impact of Non-Factorability
If the quadratic equation proves difficult or impossible to factor using standard techniques, alternative solution methods such as the quadratic formula or completing the square become necessary. Non-factorability doesn’t negate the existence of a solution; it merely necessitates employing different algebraic tools. The fact that “what is the only solution of 2x2 + 8x + x2 = 16″ is the central question presupposes a solution exists, regardless of the ease of factoring.
-
Verification through Expansion
Regardless of the factoring method used, the result can be verified by expanding the factored form to ensure it matches the original quadratic equation. For example, if factoring yields (ax + b)(cx + d), expanding this product should result in ax2 + bx + c. Successful verification confirms the accuracy of the factored form and the derived roots. In a scenario where a unique solution is sought, the expansion should unambiguously reproduce the original equation, confirming both the factoring process and the identified root.
Factoring methods, while not universally applicable, provide a direct and efficient route to solving quadratic equations when the equation’s structure lends itself to factorization. The context of seeking “what is the only solution of 2x2 + 8x + x2 = 16″ necessitates careful consideration of factorability and the potential need to employ alternative solution techniques if factoring proves challenging.
5. Unique Solution
The constraint of a “Unique Solution” fundamentally shapes the problem “what is the only solution of 2x2 + 8x + x2 = 16″. It dictates both the nature of the acceptable solution and the methods appropriate for its derivation, emphasizing the singular correct answer.
-
Discriminant Implications
For a quadratic equation in standard form (ax2 + bx + c = 0) to possess a unique solution, its discriminant (b2 – 4ac) must equal zero. This condition implies that the quadratic formula yields only one distinct root. Applying this to the simplified equation, 3x2 + 8x – 16 = 0, calculating the discriminant is crucial. If b2-4ac evaluates to zero, it confirms the potential for a unique solution and justifies focusing on methods tailored to repeated roots.
-
Repeated Root Characteristics
A unique solution in a quadratic context corresponds to a repeated root, meaning the same value satisfies the equation twice. Algebraically, this results from a perfect square trinomial after simplification. When factoring, the resulting binomial factors are identical, leading to a single solution. For instance, if factoring 3x2 + 8x – 16 = 0 yielded (3x-4)(x+4) = 0, the existence of two different roots (4/3 and -4) would invalidate the “unique solution” requirement, demanding a reevaluation of the solution process.
-
Method Validation
The claim of a unique solution necessitates stringent verification. Regardless of the solution method employed (factoring, quadratic formula, completing the square), the derived value must be substituted back into the original equation (2x2 + 8x + x2 = 16) to confirm its validity. Moreover, any intermediate steps must be examined to ensure no errors or extraneous solutions were introduced. The emphasis on “only” one solution elevates the importance of confirming the exclusivity of the identified root.
-
Real-World Constraints
In applied contexts, the “unique solution” often reflects specific physical or practical limitations. For example, if the equation modeled the dimensions of a square enclosure with a fixed area, the unique solution would represent the only possible side length. If the solution was not unique, the model or constraints would likely need revision to align with physical reality. The condition of “what is the only solution of 2x2 + 8x + x2 = 16″ guides the interpretation of the root in the real-world domain.
In summary, the “Unique Solution” condition significantly narrows the solution space and necessitates careful application and verification of algebraic techniques. It is not merely a preference for a simpler outcome but a fundamental constraint that impacts the entire problem-solving process, from discriminant analysis to method validation, influencing both mathematical accuracy and the interpretation of the solution in relevant practical applications. If a solution method yields two different results, the entire process has to be verified to identify the potential error source, so that the only solution is truly unique.
6. Algebraic Precision
Algebraic precision forms the cornerstone of accurately determining “what is the only solution of 2x2 + 8x + x2 = 16″. The stipulation of a single, unique solution intensifies the requirement for meticulous execution of each algebraic step, as any deviation from accuracy can lead to either an incorrect solution or the erroneous conclusion that no such solution exists.
-
Coefficient Handling
Accurate identification and manipulation of coefficients are paramount. Incorrectly combining like terms or misapplying the distributive property can fundamentally alter the equation, resulting in a flawed quadratic form and, consequently, an incorrect root. For instance, failing to correctly simplify 2x2 + 8x + x2 = 16 to 3x2 + 8x – 16 = 0 introduces a critical error early in the solution process. In structural engineering, where quadratic equations may model stress distribution, imprecise coefficient handling can lead to unsafe design parameters.
-
Sign Management
Maintaining precise sign conventions throughout the solution is crucial. An error in handling negative signs during rearrangement, factoring, or applying the quadratic formula can lead to a completely different solution. The constant term (-16) in 3x2 + 8x – 16 = 0 is particularly sensitive; an incorrect sign drastically alters the roots. In economic modeling, where equations might represent profit and loss scenarios, an inaccurate sign can invert the predicted outcome, leading to poor business decisions.
-
Factorization Accuracy
When employing factoring methods, the decomposition of the quadratic expression must be executed with precision. Incorrectly identifying factors or misapplying the factoring process results in inaccurate roots. If the equation were factorable as (ax + b)(cx + d) = 0, incorrect factors would yield solutions that do not satisfy the original equation. In cryptography, where algebraic manipulation underpins encryption algorithms, errors in factoring would compromise security.
-
Formulaic Application
The application of the quadratic formula requires careful substitution of coefficients and precise arithmetic operations. Incorrectly substituting values or making calculation errors, particularly under the square root, invalidates the result. The quadratic formula’s reliance on ‘a’, ‘b’, and ‘c’ necessitates their accurate identification. In signal processing, where quadratic equations might model filter responses, errors in formula application could lead to distorted or unusable signals.
In conclusion, algebraic precision is not merely a desirable attribute, but an essential requirement for solving “what is the only solution of 2x2 + 8x + x2 = 16″ with certainty. Each facet, from coefficient handling to formulaic application, demands meticulous attention to detail. The stipulation of a unique solution amplifies the consequences of even minor errors, emphasizing the critical role of accuracy in obtaining the correct root and confirming its exclusivity.
7. Verification Process
The verification process is an indispensable component in the context of “what is the only solution of 2x2 + 8x + x2 = 16″. Given the constraint of a single, unique solution, rigorous verification becomes crucial to confirm the correctness and exclusivity of the derived root. The process ensures that the obtained value for ‘x’ not only satisfies the equation but also that no other value does so, validating the entire problem-solving approach.
-
Substitution into Original Equation
The primary verification step involves substituting the derived solution back into the original equation: 2x2 + 8x + x2 = 16. If the derived value for ‘x’ correctly balances the equation, demonstrating that both sides are equal, it provides initial validation. This is particularly important in contexts such as projectile motion calculations, where an incorrectly verified solution could lead to inaccurate targeting predictions. This substitution must use the original equation before simplification, as that mitigates errors introduced during simplification.
-
Discriminant Confirmation
Given the constraint of a single solution, the discriminant (b2 – 4ac) of the simplified quadratic equation (3x2 + 8x – 16 = 0) should evaluate to zero. Calculating the discriminant provides an independent confirmation of the solution’s nature. A non-zero discriminant invalidates the “only solution” condition and signals an error in the prior steps. Consider applications in electrical circuit analysis, where quadratic equations might model impedance; a discriminant verification ensures that the derived impedance value corresponds to the expected behavior of the circuit.
-
Factorization Validation
If factoring is the chosen method, expanding the factored form must precisely reproduce the original quadratic expression. Any deviation indicates an error in the factorization process or the identification of the roots. Expanding the potential factors demonstrates that they are equivalent to the original simplified formula. Accurate factorization is relevant in various fields, such as optimization problems, where incorrectly factored equations would lead to suboptimal solutions.
-
Extraneous Solution Elimination
Although the problem stipulates a single solution, it is still necessary to check for extraneous solutions, particularly if any intermediate steps involved squaring both sides of an equation or other operations that could introduce spurious roots. Extraneous solutions are values that emerge during the solving process but do not satisfy the original equation. If this occurred, the verification process should identify these cases for elimination. In the context of acoustics where equations model sound wave propagation, the unique real solution often represent a specific frequency or amplitude.
The verification process in “what is the only solution of 2x2 + 8x + x2 = 16″ is not merely a procedural step, but an integral component of the solution itself. It ensures that the identified root is mathematically sound and aligns with the problem’s specific constraints. By systematically validating each step, potential errors are identified and corrected, reinforcing the integrity of the final result and confirming that the derived value for ‘x’ is indeed the only valid solution.
8. Standard Form
The expression “Standard Form” is intrinsically linked to the task of determining “what is the only solution of 2x2 + 8x + x2 = 16.” Transformation into standard form is often a necessary precursor to applying solution methodologies for quadratic equations, enabling accurate coefficient identification and the subsequent application of appropriate techniques.
-
Equation Simplification and Rearrangement
The initial equation, 2x2 + 8x + x2 = 16, is not in standard quadratic form. The first step towards finding a solution involves simplifying and rearranging the terms to conform to the standard structure: ax2 + bx + c = 0. In this specific case, the equation transforms to 3x2 + 8x – 16 = 0. This simplification and rearrangement are crucial because standard solution methods, such as factoring or applying the quadratic formula, are designed to operate on equations in this precise form. Without accurate conversion to standard form, subsequent calculations become unreliable and lead to incorrect solutions. For instance, in physics, if ‘x’ represented time, an inaccurate ‘x’ value would result in a wrong physical outcome.
-
Coefficient Identification and Application
Standard form directly facilitates the identification of the coefficients a, b, and c. In the simplified equation, 3x2 + 8x – 16 = 0, a = 3, b = 8, and c = -16. These coefficients are essential inputs for solving the quadratic equation using methods like the quadratic formula. Incorrectly identifying the coefficients, due to a failure to arrange the equation in standard form, leads to errors in applying these solution techniques, rendering the resulting roots invalid. In economic models, these coefficients are parameters of an economic system; incorrect values would lead to inaccurate predictions.
-
Discriminant Analysis and Solution Nature
The discriminant, b2 – 4ac, which determines the nature of the solutions to a quadratic equation, is directly calculated using the coefficients obtained from the standard form. In the equation 3x2 + 8x – 16 = 0, the discriminant is 82 – 4(3)(-16) = 64 + 192 = 256. If the discriminant is zero, there is one real repeated root. If it is positive, there are two distinct real roots. If it is negative, there are two complex roots. Knowing the discriminant is a prerequisite to selecting the appropriate method. In structural engineering, these solutions might represent values of force necessary for mechanical equilibrium, and if the solution is complex, the structure is not in equilibrium and has failed.
The transformation of 2x2 + 8x + x2 = 16 into standard form is not merely a stylistic preference; it is a foundational step that enables the application of standard solution techniques and guarantees the accurate determination of the root(s). Neglecting this step compromises the entire problem-solving process, emphasizing the inherent link between standard form and “what is the only solution.” Failure to adhere to the rules for algebraic manipulation during transformation would fundamentally alter the equation being solved, resulting in an entirely different problem with potentially unrelated solutions. Understanding the role of standard form is, therefore, crucial for anyone attempting to solve quadratic equations accurately and effectively.
9. Equation Simplification
Equation simplification is a critical prerequisite to determining “what is the only solution of 2x2 + 8x + x2 = 16.” The given equation, in its initial form, presents a degree of complexity that obscures the direct application of standard solution methods for quadratic equations. Simplification acts as the enabling step that transforms the expression into a manageable form, facilitating the identification of coefficients and the subsequent application of solution techniques. The act of simplifying involves combining like terms and rearranging the equation to fit the standard quadratic structure: ax2 + bx + c = 0. For this specific case, the equation 2x2 + 8x + x2 = 16 must be simplified to 3x2 + 8x – 16 = 0. This step is not merely a cosmetic change; it is a fundamental transformation that determines the validity of subsequent calculations. Without accurate simplification, any attempt to solve the equation will inevitably lead to an incorrect result. Real-world scenarios, such as modelling the trajectory of a projectile, demand precise simplification, and incorrect simplification results in missing the target by a wide margin.
The simplification process directly influences the accuracy of coefficient identification, a factor essential for applying the quadratic formula or factoring techniques. Misidentification of the coefficients, stemming from a failure to simplify correctly, propagates errors throughout the entire solution process. For example, treating the original equation as separate terms without combining the ‘x2‘ terms would lead to an incorrect assessment of the ‘a’ coefficient, compromising the accuracy of both the discriminant calculation and the derived roots. Furthermore, the simplification reveals the true nature of the quadratic equation, determining whether it possesses two distinct real roots, a single repeated root, or complex roots. Consider the field of financial modeling; if the equation described the breakeven point of a business, errors in simplification could result in incorrect assessments of profitability and solvency, leading to poor financial decisions.
In summary, equation simplification is not merely a preliminary step but an integral component of finding “what is the only solution of 2x2 + 8x + x2 = 16″. It directly impacts the accuracy of coefficient identification, influences the choice of appropriate solution methods, and ensures the validity of the final root. Neglecting or performing this step incorrectly compromises the entire problem-solving process and can have significant consequences in real-world applications where quadratic equations serve as valuable models. Thus, simplification not only guides the determination of solution, but also serves as a filtering procedure that provides the path to the sole correct answer and discarding others.
Frequently Asked Questions about the Solution to 2x2 + 8x + x2 = 16
The following questions address common inquiries and potential misunderstandings related to solving the equation 2x2 + 8x + x2 = 16. These questions aim to clarify the process and ensure a thorough understanding of the solution.
Question 1: Why is simplification necessary before solving the equation?
Simplification combines like terms and rearranges the equation into the standard quadratic form (ax2 + bx + c = 0), which is essential for applying standard solution methods, such as factoring or the quadratic formula. Without simplification, accurate coefficient identification is impossible, leading to incorrect results.
Question 2: What does it mean for a quadratic equation to have only one solution?
A single solution indicates a repeated real root, implying the discriminant (b2 – 4ac) of the quadratic equation is equal to zero. This signifies that the graph of the quadratic function touches the x-axis at only one point.
Question 3: How does coefficient identification impact the solution process?
Correctly identifying the coefficients (a, b, and c) is crucial for both factoring and applying the quadratic formula. Accurate coefficients are essential for correctly finding the roots. Errors in coefficient identification propagate through the rest of the calculation, leading to incorrect solutions.
Question 4: If factoring is difficult, are there alternative solution methods?
Yes. If factoring is challenging or impossible, alternative methods like the quadratic formula or completing the square can be used. These methods guarantee a solution even when factoring is not straightforward.
Question 5: How is the solution verified to ensure its accuracy?
Verification involves substituting the derived solution back into the original equation to confirm that it balances the equation. Also, double-checking the discriminant calculation ensures that the type and number of solutions have been predicted.
Question 6: What is the significance of the discriminant in determining the nature of the solution?
The discriminant (b2 – 4ac) provides information about the number and type of solutions. A discriminant of zero indicates one real repeated root, a positive discriminant indicates two distinct real roots, and a negative discriminant indicates two complex roots.
Understanding these frequently asked questions is essential for effectively addressing the problem of finding the single solution for the given equation. Paying attention to detail and verifying the solution ensures a robust solution to the problem.
The following section will delve into detailed examples to illustrate methods of solving 2x2 + 8x + x2 = 16.
Tips for Determining the Unique Solution of Quadratic Equations
This section offers practical guidance for efficiently and accurately solving quadratic equations that possess a single, unique solution.
Tip 1: Prioritize Simplification. Before applying any solution method, simplify the equation by combining like terms and rearranging it into the standard quadratic form (ax2 + bx + c = 0). This step ensures accurate coefficient identification and reduces the risk of algebraic errors. For example, transform 2x2 + 8x + x2 = 16 into 3x2 + 8x – 16 = 0 before proceeding.
Tip 2: Analyze the Discriminant. Calculate the discriminant (b2 – 4ac) early in the process. If the discriminant is zero, it confirms that the equation has only one real root, thereby streamlining the solution process and focusing efforts on methods suited for repeated roots. A non-zero discriminant indicates either an error or that the problem constraints are not being followed. In the given problem, a discriminant of 256 indicates there are two solutions, and that this conflicts with the only one solution constraint.
Tip 3: Verify Coefficient Accuracy. Double-check the identified coefficients (a, b, and c) before applying the quadratic formula or attempting factorization. Incorrect coefficients lead to incorrect roots. The values must adhere to accurate transformation of the original formula, because errors at that step would yield incorrect roots.
Tip 4: Factor Strategically. When factoring, focus on finding factors that lead to a repeated root. This often involves identifying perfect square trinomials or binomial expressions that are identical. The aim is to ensure both factors of x resolve to the same single solution. The factors of the original formula can then be checked for accuracy by expansion.
Tip 5: Substitute for Validation. After obtaining a potential solution, always substitute it back into the original equation (before simplification) to verify its accuracy. This step catches errors made during simplification or the solution process, ensuring the validity of the result. If you solve for one answer, and there should be two answers, then the substitution step will lead to errors.
Tip 6: Employ Alternative Methods. If factoring proves difficult, consider using the quadratic formula or completing the square. These methods provide reliable solutions regardless of the equation’s factorability. Using the quadratic equation, it can be shown for the original formula that x = (-8 + sqrt(256)) / 6 or x = (-8 – sqrt(256)) / 6, which means x = 4/3 or x= -4. These does not align with a unique solution, so there is likely an error. Thus this approach demonstrates the power of using multiple approaches.
Adhering to these tips ensures a more efficient and accurate process for solving quadratic equations with unique solutions, minimizing errors and promoting a deeper understanding of the underlying mathematical principles.
The subsequent segment will offer a comprehensive conclusion summarizing the key findings and insights derived throughout this exposition.
Conclusion
The preceding analysis meticulously explored the problem of determining “what is the only solution of 2x2 + 8x + x2 = 16.” Through detailed examination of standard form transformation, coefficient identification, solution methodologies, and verification procedures, it becomes apparent that the equation, upon correct simplification to 3x2 + 8x – 16 = 0, possesses two distinct real roots, thus contradicting the initial premise of a single, unique solution. The discrepancy between the problem’s assertion and the mathematical reality highlights the critical importance of rigorous algebraic manipulation and discriminant analysis in solving quadratic equations.
This exploration underscores the need for careful scrutiny of problem statements and a commitment to mathematical accuracy. While the initial framing of the problem posited a singular solution, the analytical process revealed a more complex reality. This emphasizes the necessity for continued engagement with mathematical principles and a commitment to evidence-based reasoning in approaching problem-solving across various domains. Further investigation into the source of the problem’s assertion may reveal insights into the nature of mathematical errors or the influence of contextual factors on solution interpretation.
