Finding The Value Of C For A Perfect Square Quadratic Expression

Introduction

In this article, we will explore how to determine the value of c that transforms the quadratic expression ${2x^2 - 10x + 5 + c}$ into a perfect square. Understanding this concept is crucial in various areas of algebra, including solving quadratic equations, graphing parabolas, and simplifying complex expressions. This article aims to provide a comprehensive guide, ensuring clarity and thorough understanding for students and enthusiasts alike. Let's dive into the methods and techniques needed to solve this interesting problem.

Understanding Perfect Square Trinomials

Perfect square trinomials are quadratic expressions that can be factored into the form ${(ax + b)^2}$ or ${(ax - b)^2}$. Expanding these forms gives us ${a^2x^2 + 2abx + b^2}$ and ${a^2x^2 - 2abx + b^2}$ respectively. Identifying a perfect square trinomial involves recognizing the relationship between the coefficients of the quadratic term, the linear term, and the constant term. Specifically, the constant term should be the square of half the coefficient of the linear term (after factoring out the coefficient of the quadratic term if it's not 1). For instance, in the trinomial ${x^2 + 6x + 9}$, the constant term 9 is the square of half the coefficient of x (which is 6/2 = 3, and ${3^2 = 9}$). Recognizing these patterns is crucial for solving problems related to perfect squares.

When we talk about perfect square trinomials, it's essential to understand their algebraic structure. A perfect square trinomial is a trinomial that results from squaring a binomial. The general form of a perfect square trinomial is ${a^2x^2 + 2abx + b^2}$ or ${a^2x^2 - 2abx + b^2}$. These trinomials can be factored into ${(ax + b)^2}$ and ${(ax - b)^2}$ respectively. Recognizing these forms is critical in algebra for simplifying expressions, solving equations, and various other applications. For a quadratic expression to be a perfect square, the constant term must be precisely the value that completes the square. This value is derived from the coefficient of the linear term and the coefficient of the quadratic term. This is particularly useful in solving quadratic equations and simplifying algebraic expressions.

To further illustrate, consider the trinomial ${4x^2 + 12x + 9}$. Here, ${a^2 = 4}$, so ${a = 2}$. The linear term is ${12x}$, which should match ${2abx}$. Since ${a = 2}$, we have ${2 * 2 * b = 12}$, which gives ${b = 3}$. The constant term is 9, which is indeed ${b^2 = 3^2}$. Therefore, ${4x^2 + 12x + 9}$ is a perfect square trinomial and can be factored as ${(2x + 3)^2}$. Understanding these patterns and being able to quickly identify perfect square trinomials can significantly simplify algebraic manipulations and problem-solving. This skill is invaluable not only in algebra but also in calculus and other advanced mathematical fields. The ability to recognize and manipulate perfect square trinomials allows for more efficient solutions and a deeper understanding of algebraic structures.

Completing the Square

The method of completing the square is a powerful technique used to rewrite a quadratic expression in a form that reveals its perfect square component. This method is particularly useful when the quadratic expression is not immediately recognizable as a perfect square trinomial. The general idea is to manipulate the expression by adding and subtracting a constant term, effectively creating a perfect square trinomial plus a constant. This technique is widely used in solving quadratic equations, deriving the quadratic formula, and graphing quadratic functions. The core principle involves transforming the quadratic expression into the form ${a(x + h)^2 + k}$, where ${(x + h)^2}$ represents the perfect square part and ${k}$ is the constant term.

To complete the square, consider a quadratic expression in the form ${ax^2 + bx + c}$. The first step is to factor out the coefficient a from the quadratic and linear terms, if a is not equal to 1. This gives us ${a(x^2 + (b/a)x) + c}$. Next, we focus on the expression inside the parentheses. To complete the square, we need to add and subtract a value that makes the expression inside the parentheses a perfect square. This value is obtained by taking half of the coefficient of x (which is ${b/a}$), squaring it, and both adding and subtracting it inside the parentheses. So, we add and subtract ${(b/2a)^2}$. This leads to the expression ${a[x^2 + (b/a)x + (b/2a)^2 - (b/2a)^2] + c}$. The first three terms inside the brackets now form a perfect square, which can be written as ${(x + b/2a)^2}$. The expression then becomes ${a[(x + b/2a)^2 - (b/2a)^2] + c}$. Finally, we distribute a and simplify to get the completed square form ${a(x + b/2a)^2 + c - a(b/2a)^2}$. This form clearly shows the perfect square component and the constant term, making it easier to analyze and solve quadratic equations.

For example, let’s complete the square for the expression ${x^2 + 6x + 5}$. The coefficient of ${x^2}$ is 1, so we don’t need to factor anything out. We take half of the coefficient of x, which is 6/2 = 3, and square it to get ${3^2 = 9}$. We add and subtract 9 inside the expression: ${x^2 + 6x + 9 - 9 + 5}$. The first three terms form a perfect square: ${(x + 3)^2}$. So the expression becomes ${(x + 3)^2 - 9 + 5}$, which simplifies to ${(x + 3)^2 - 4}$. This is the completed square form, and it clearly shows that the vertex of the parabola represented by this quadratic is at (-3, -4). The technique of completing the square is not just a method for solving equations; it is a fundamental algebraic tool that provides deep insights into the structure and properties of quadratic expressions.

Problem Analysis: 2x² - 10x + 5 + c

To find the value of c for which the expression ${2x^2 - 10x + 5 + c}$ is a perfect square, we need to apply the method of completing the square. This involves manipulating the given expression to fit the form of a perfect square trinomial. The expression should be transformed into a form ${(ax + b)^2}$ or ${(ax - b)^2}$ for some constants a and b. By comparing the coefficients, we can determine the value of c that makes the expression a perfect square. The key here is to ensure that the constant term, after completing the square, is such that the entire expression can be written as the square of a binomial. This process involves careful algebraic manipulation and a solid understanding of the structure of perfect square trinomials.

The first step in analyzing the given expression ${2x^2 - 10x + 5 + c}$ is to factor out the coefficient of the ${x^2}$ term, which is 2. Factoring out 2 from the first two terms gives us ${2(x^2 - 5x) + 5 + c}$. Now, we need to complete the square for the expression inside the parentheses. To do this, we take half of the coefficient of x, which is -5, and square it. Half of -5 is -5/2, and squaring it gives us ${(-5/2)^2 = 25/4}$. We add and subtract this value inside the parentheses: ${2(x^2 - 5x + 25/4 - 25/4) + 5 + c}$. The first three terms inside the parentheses now form a perfect square: ${x^2 - 5x + 25/4 = (x - 5/2)^2}$. So the expression becomes ${2[(x - 5/2)^2 - 25/4] + 5 + c}$. Distributing the 2, we get ${2(x - 5/2)^2 - 2(25/4) + 5 + c}$, which simplifies to ${2(x - 5/2)^2 - 25/2 + 5 + c}$.

Now, we simplify the constant terms: ${-25/2 + 5 + c = -25/2 + 10/2 + c = -15/2 + c}$. The expression is now in the form ${2(x - 5/2)^2 - 15/2 + c}$. For the original expression to be a perfect square, the constant term must be zero. This is because a perfect square trinomial has the form ${(ax + b)^2}$, which, when expanded, has no additional constant term separate from the squared binomial. Therefore, we set the constant term equal to zero: ${-15/2 + c = 0}$. Solving for c, we get ${c = 15/2}$. This means that when ${c = 15/2}$, the expression ${2x^2 - 10x + 5 + c}$ becomes a perfect square. Thus, by completing the square and setting the constant term to zero, we have successfully found the value of c that satisfies the given condition. This method provides a clear and systematic approach to solving such problems, ensuring accuracy and a deep understanding of the underlying algebraic principles.

Solving for c

To solve for c, we need to set the constant term in the completed square form of the expression equal to zero. This is because, for the quadratic expression to be a perfect square, the remaining term must be a perfect square trinomial, which implies that the additional constant term should be zero. From our previous analysis, we have the expression in the form ${2(x - 5/2)^2 - 15/2 + c}$. The constant term here is ${-15/2 + c}$. Setting this equal to zero allows us to isolate and find the value of c.

We set the constant term to zero: ${-15/2 + c = 0}$. To solve for c, we simply add 15/2 to both sides of the equation: ${c = 15/2}$. This means that when ${c = 15/2}$, the expression ${2x^2 - 10x + 5 + c}$ becomes a perfect square. To verify this, we can substitute ${c = 15/2}$ back into the original expression: ${2x^2 - 10x + 5 + 15/2}$. We can rewrite 5 as 10/2, so the expression becomes ${2x^2 - 10x + 10/2 + 15/2}$, which simplifies to ${2x^2 - 10x + 25/2}$. Factoring out 2, we get ${2(x^2 - 5x + 25/4)}$. The expression inside the parentheses is indeed a perfect square: ${(x - 5/2)^2}$. Thus, the entire expression becomes ${2(x - 5/2)^2}$, which is a perfect square.

Therefore, the value of c that makes the given quadratic expression a perfect square is ${15/2}$. This solution is obtained by completing the square, identifying the constant term, and setting it to zero. This method not only provides the value of c but also confirms that the resulting expression is indeed a perfect square. Understanding and applying this technique is essential for solving various problems in algebra, particularly those involving quadratic expressions and perfect square trinomials. The ability to manipulate and analyze quadratic expressions is a fundamental skill in mathematics, enabling the solution of complex problems across different branches of the field.

Conclusion

In conclusion, we have successfully determined the value of c that makes the quadratic expression ${2x^2 - 10x + 5 + c}$ a perfect square. By employing the method of completing the square and setting the constant term to zero, we found that ${c = 15/2}$. This process not only provides the solution but also reinforces the understanding of perfect square trinomials and their properties. The ability to manipulate quadratic expressions and identify perfect squares is a valuable skill in algebra and beyond.

This exploration demonstrates the importance of understanding algebraic techniques such as completing the square. These methods are not only useful for solving specific problems but also provide a deeper insight into the structure and behavior of quadratic expressions. The process of completing the square, in particular, is a powerful tool with applications in various areas of mathematics, including solving equations, graphing functions, and simplifying expressions. Mastering these techniques is essential for students and professionals alike, enabling them to tackle more complex problems with confidence and precision.

Ultimately, the problem of finding the value of c highlights the interconnectedness of different concepts in algebra. The ability to recognize perfect square trinomials, complete the square, and solve equations are all crucial skills that build upon each other. By working through this problem, we have not only found a specific solution but also strengthened our understanding of these fundamental concepts. This comprehensive approach is key to success in mathematics and demonstrates the value of methodical problem-solving techniques.