Solving $x^2=12x-15$ By Completing The Square A Step-by-Step Guide

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Introduction

In this comprehensive article, we will delve into the method of completing the square to solve the quadratic equation $x^2 = 12x - 15$. This technique is a powerful algebraic tool that allows us to rewrite any quadratic equation in a form that makes it easy to find the solutions. By the end of this guide, you will have a clear understanding of how to apply this method and accurately determine the solution set for the given equation. Quadratic equations are fundamental in mathematics, appearing in various fields such as physics, engineering, and economics. Mastering the technique of completing the square provides a solid foundation for solving more complex mathematical problems.

The importance of solving quadratic equations cannot be overstated. They arise in numerous real-world applications, from modeling projectile motion to designing electrical circuits. The method of completing the square is particularly valuable because it not only provides the solutions but also gives insight into the structure of the quadratic equation itself. It allows us to rewrite the equation in vertex form, which reveals the vertex of the parabola represented by the equation. This makes completing the square a versatile tool in both theoretical and practical contexts.

Understanding the Basics of Quadratic Equations

Before we dive into the step-by-step solution, it's crucial to understand the basic form of a quadratic equation and the logic behind completing the square. A quadratic equation is typically expressed in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable we want to solve for. The method of completing the square involves transforming this general form into a perfect square trinomial, which can then be easily factored. This transformation allows us to isolate $x$ and find its values.

The solution set of a quadratic equation consists of the values of $x$ that satisfy the equation. These values are also known as the roots or zeros of the quadratic equation. A quadratic equation can have two distinct real roots, one repeated real root, or two complex roots, depending on the discriminant ($b^2 - 4ac$) of the equation. Completing the square is a reliable method for finding these roots, regardless of their nature. This technique provides a systematic approach to solving quadratic equations, offering a clear and concise way to determine the solution set.

Step-by-Step Solution

Step 1: Rewrite the Equation

First, we need to rewrite the given equation $x^2 = 12x - 15$ in the standard quadratic form, which is $ax^2 + bx + c = 0$. To do this, we subtract $12x$ and add $15$ to both sides of the equation:

$x^2 - 12x + 15 = 0$

This rearrangement sets the stage for completing the square. By having all terms on one side and zero on the other, we can manipulate the equation to form a perfect square trinomial. This step is crucial because it aligns the equation with the structure required for applying the completing the square method.

Step 2: Complete the Square

To complete the square, we need to add and subtract a value that will make the left side of the equation a perfect square trinomial. A perfect square trinomial is a quadratic expression that can be factored into the form $(x + k)^2$ or $(x - k)^2$, where $k$ is a constant. The value we need to add and subtract is determined by taking half of the coefficient of the $x$ term (which is $-12$ in this case), squaring it, and adding and subtracting the result. Half of $-12$ is $-6$, and $(-6)^2$ is $36$.

So, we add and subtract $36$ to the left side of the equation:

$x^2 - 12x + 36 - 36 + 15 = 0$

This step is the heart of the method of completing the square. By adding and subtracting the appropriate value, we maintain the equality of the equation while creating a perfect square trinomial. This trinomial can then be factored, which simplifies the equation and allows us to solve for $x$.

Step 3: Factor the Perfect Square Trinomial

Now, we can factor the perfect square trinomial $x^2 - 12x + 36$. This trinomial can be factored as $(x - 6)^2$. So, the equation becomes:

$(x - 6)^2 - 36 + 15 = 0$

Simplifying the constant terms, we get:

$(x - 6)^2 - 21 = 0$

This factorization is a key step in the process. It transforms the quadratic expression into a more manageable form, where we have a squared term and a constant. This form allows us to easily isolate the variable $x$ and find its values.

Step 4: Isolate the Squared Term

Next, we isolate the squared term by adding $21$ to both sides of the equation:

$(x - 6)^2 = 21$

This isolation is essential because it sets up the equation for taking the square root. By isolating the squared term, we can eliminate the square and solve for $x$ in the next step.

Step 5: Take the Square Root

Now, we take the square root of both sides of the equation. Remember to consider both the positive and negative square roots:

$x - 6 = \{ ext{\sqrt{21}}, -\sqrt{21} \}$

This step is crucial for finding both solutions of the quadratic equation. By considering both the positive and negative square roots, we ensure that we capture all possible values of $x$ that satisfy the equation. This is a common point of error, so it's important to remember to include both roots.

Step 6: Solve for x

Finally, we solve for $x$ by adding $6$ to both sides of the equation:

$x = 6 \{+\sqrt{21}, -\sqrt{21}\}$

This gives us two solutions:

$x = 6 + \sqrt{21}$

$x = 6 - \sqrt{21}$

Therefore, the solution set is $\{6 - \sqrt{21}, 6 + \sqrt{21}\}$. This is the final step in the process, where we isolate $x$ and find its values. The solution set represents the roots of the quadratic equation, which are the values of $x$ that make the equation true.

Verifying the Solution

To ensure our solution is correct, we can substitute the values of $x$ back into the original equation $x^2 = 12x - 15$ and check if the equation holds true.

Verification for $x = 6 + \sqrt{21}$

Substituting $x = 6 + \sqrt{21}$ into the equation:

$(6 + \sqrt{21})^2 = 12(6 + \sqrt{21}) - 15$

Expanding the left side:

$36 + 12\sqrt{21} + 21 = 72 + 12\sqrt{21} - 15$

Simplifying:

$57 + 12\sqrt{21} = 57 + 12\sqrt{21}$

This is true, so $x = 6 + \sqrt{21}$ is a valid solution.

Verification for $x = 6 - \sqrt{21}$

Substituting $x = 6 - \sqrt{21}$ into the equation:

$(6 - \sqrt{21})^2 = 12(6 - \sqrt{21}) - 15$

Expanding the left side:

$36 - 12\sqrt{21} + 21 = 72 - 12\sqrt{21} - 15$

Simplifying:

$57 - 12\sqrt{21} = 57 - 12\sqrt{21}$

This is also true, so $x = 6 - \sqrt{21}$ is a valid solution.

By verifying both solutions, we can be confident that our answer is correct. This step is a crucial part of the problem-solving process, as it helps to catch any errors that may have occurred during the calculations.

Conclusion

In this article, we have successfully solved the quadratic equation $x^2 = 12x - 15$ by completing the square. We found that the solution set is $\{6 - \sqrt{21}, 6 + \sqrt{21}\}$. The method of completing the square is a powerful technique for solving quadratic equations, providing a systematic approach to finding the roots. It is applicable to any quadratic equation and offers a clear and concise way to determine the solution set.

Understanding and mastering this method is essential for anyone studying algebra and beyond. It provides a solid foundation for solving more complex mathematical problems and is a valuable tool in various fields of science and engineering. By following the step-by-step process outlined in this article, you can confidently solve quadratic equations by completing the square and verify your solutions to ensure accuracy.