Solving $(x+7)(x-9)=25$ By Completing The Square Method

In mathematics, solving quadratic equations is a fundamental skill. One powerful technique for finding the solutions (also known as roots) of a quadratic equation is completing the square. This method is particularly useful when the equation is not easily factorable or when you need to rewrite the equation in vertex form. This article will explain completing the square in detail and guide you through the steps. We will use the example equation $(x+7)(x-9)=25$ and solve for $x$ by completing the square.

Understanding Quadratic Equations

Before diving into completing the square, let's understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, generally written in the form:

$ax^2 + bx + c = 0$

where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. The solutions (or roots) of a quadratic equation are the values of $x$ that satisfy the equation. These solutions can be real or complex numbers. Quadratic equations have a wide range of applications in various fields, including physics, engineering, economics, and computer science.

The equation $(x+7)(x-9)=25$ is a quadratic equation presented in a factored form. To solve it, we first need to expand it into the standard form $ax^2 + bx + c = 0$. This involves multiplying the binomials and simplifying the expression. Once we have the equation in standard form, we can then proceed with the method of completing the square. The process of expanding and simplifying the equation is crucial because it sets the stage for applying the completing the square technique effectively. By understanding the structure of quadratic equations and how to manipulate them, you gain a solid foundation for solving a variety of mathematical problems.

The Method of Completing the Square

Completing the square is a technique used to convert a quadratic equation from the standard form $ax^2 + bx + c = 0$ into the vertex form $a(x - h)^2 + k = 0$, where $(h, k)$ is the vertex of the parabola represented by the quadratic equation. This method is particularly useful because it allows us to easily identify the vertex of the parabola and solve for the roots of the equation. The vertex form provides a clear picture of the parabola's symmetry and its minimum or maximum value, depending on the sign of $a$. Understanding the geometric interpretation of completing the square can enhance your problem-solving skills and provide a deeper insight into quadratic functions.

The main idea behind completing the square is to manipulate the quadratic expression to form a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(x + p)^2$ or $(x - p)^2$. This manipulation involves adding and subtracting a specific value to the equation, which allows us to rewrite the quadratic expression as a squared binomial plus a constant. The process requires careful attention to algebraic details and a good understanding of factoring techniques. By mastering completing the square, you can handle a wide range of quadratic equations and transformations, making it a valuable tool in algebraic problem-solving.

Steps to Solve $(x+7)(x-9)=25$ by Completing the Square

Let's walk through the steps to solve the equation $(x+7)(x-9)=25$ using completing the square.

Step 1: Expand and Simplify the Equation

First, we need to expand the left side of the equation:

$(x+7)(x-9) = x^2 - 9x + 7x - 63$

Combine like terms:

$x^2 - 2x - 63 = 25$

Next, subtract 25 from both sides to set the equation to zero:

$x^2 - 2x - 63 - 25 = 0$

$x^2 - 2x - 88 = 0$

The initial expansion and simplification are critical for transforming the given equation into the standard quadratic form. This form allows us to easily identify the coefficients needed for completing the square. By accurately expanding and combining like terms, we avoid errors that could complicate the subsequent steps. Setting the equation to zero is a necessary step for applying the completing the square method, as it allows us to isolate the quadratic expression and manipulate it effectively. This foundational step ensures that we can proceed with the rest of the process smoothly and accurately.

Step 2: Move the Constant Term to the Right Side

Move the constant term (-88) to the right side of the equation:

$x^2 - 2x = 88$

Isolating the $x^2$ and $x$ terms on one side of the equation is a crucial step in the process of completing the square. This separation allows us to focus on the terms that will form the perfect square trinomial. By moving the constant term to the other side, we create a space to add the value needed to complete the square without altering the balance of the equation. This step is not merely an algebraic manipulation but a strategic move that sets the stage for rewriting the quadratic expression in a more manageable form. Accurate isolation of these terms ensures that the subsequent steps lead to a correct solution.

Step 3: Complete the Square

To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. The value we need to add is $(\frac{b}{2})^2$, where $b$ is the coefficient of the $x$ term. In our equation, $b = -2$.

Calculate the value to add:

$(\frac{-2}{2})^2 = (-1)^2 = 1$

Add 1 to both sides of the equation:

$x^2 - 2x + 1 = 88 + 1$

$x^2 - 2x + 1 = 89$

Completing the square is the heart of this method. The formula $(\frac{b}{2})^2$ provides the precise value needed to create a perfect square trinomial. This step transforms the quadratic expression into a form that can be easily factored. Adding the same value to both sides of the equation maintains the balance and ensures the integrity of the solution. The resulting perfect square trinomial is a key element in simplifying the equation and solving for $x$. This meticulous calculation and addition are essential for accurately applying the completing the square technique.

Step 4: Factor the Left Side

The left side is now a perfect square trinomial, which can be factored as:

$(x - 1)^2 = 89$

Factoring the perfect square trinomial is a pivotal step as it simplifies the equation into a more manageable form. The factored form $(x - 1)^2$ clearly shows the square of a binomial, which allows us to easily take the square root in the next step. This simplification is a direct result of completing the square and transforms the quadratic expression into a format that is straightforward to solve. Accurate factoring here is crucial for the subsequent steps and leads us closer to finding the roots of the equation.

Step 5: Take the Square Root of Both Sides

Take the square root of both sides of the equation:

$\sqrt{(x - 1)^2} = \pm\sqrt{89}$

$x - 1 = \pm\sqrt{89}$

Taking the square root of both sides is a critical step that allows us to isolate the variable $x$. It is essential to consider both the positive and negative square roots, as both will satisfy the equation. This step directly follows from the factored form achieved by completing the square and leads us to a simple linear equation. Remembering to include both the positive and negative roots ensures that we find all possible solutions for $x$.

Step 6: Solve for $x$

Add 1 to both sides to solve for $x$:

$x = 1 \pm \sqrt{89}$

The final step in solving for $x$ involves isolating $x$ by adding 1 to both sides of the equation. This step provides the exact solutions to the quadratic equation in terms of a constant and a square root. The two solutions, $1 + \sqrt{89}$ and $1 - \sqrt{89}$, represent the points where the parabola intersects the x-axis. This step completes the process of completing the square and yields the roots of the original equation. These values are the definitive solutions and can be used for further analysis or application.

Final Answer

The solutions for $x$ are:

$x = 1 + \sqrt{89}$ or $x = 1 - \sqrt{89}$

Therefore, the correct answer is:

C. $x=1 \pm \sqrt{89}$

Conclusion

Completing the square is a valuable technique for solving quadratic equations, especially when factoring is not straightforward. By following these steps, you can confidently solve any quadratic equation using this method. Remember to practice and review these steps to master the technique. Understanding and applying completing the square enhances your problem-solving skills and provides a solid foundation for more advanced mathematical concepts.