Solve Quadratic Equations By Completing The Square

Let's dive into solving the quadratic equation ${x^2 + 12x + 25 = 0}$ by completing the square. This method is super useful when the equation doesn't factor nicely. We'll break it down step by step so it’s easy to follow. By the end of this guide, you'll be a pro at completing the square!

Understanding Completing the Square

Completing the square is a technique used to rewrite a quadratic equation in a form that allows us to easily solve for ${x}$. The goal is to transform the equation into the form ${(x + a)^2 = b}$, where ${a}$ and ${b}$ are constants. This makes it straightforward to isolate ${x}$ and find its values. It's especially handy when the quadratic equation can't be easily factored. The main idea behind completing the square is to manipulate the quadratic expression to create a perfect square trinomial on one side of the equation. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as ${(x + a)^2}$ or ${(x - a)^2}$. To complete the square, we need to add and subtract a specific value to the quadratic expression. This value is determined by taking half of the coefficient of the ${x}$ term, squaring it, and then adding and subtracting it within the equation. This process ensures that we maintain the equality of the equation while creating a perfect square trinomial. Once we have completed the square, we can rewrite the quadratic equation in the form ${(x + a)^2 = b}$. From there, we can easily solve for ${x}$ by taking the square root of both sides of the equation and isolating ${x}$. Remember, when taking the square root, we need to consider both the positive and negative roots, as both will satisfy the equation. Completing the square is a versatile technique that can be applied to any quadratic equation, regardless of whether it can be factored easily or not. It provides a systematic approach to solving quadratic equations and is particularly useful when dealing with equations that have irrational or complex roots. Mastering this technique will significantly enhance your ability to solve quadratic equations and tackle more advanced mathematical problems. Keep practicing, and you'll become more comfortable and confident in applying completing the square to various types of quadratic equations.

Step-by-Step Solution

(a) Rewriting the Equation

First, let's rewrite the given equation ${x^2 + 12x + 25 = 0}$. Our goal is to get it into the form ${(x + a)^2 = b}$.

1. Move the constant term to the right side: Subtract 25 from both sides of the equation: ${x^2 + 12x = -25}$

2. Complete the square: To complete the square, we need to add and subtract ${(\frac{12}{2})^2 = 6^2 = 36}$ to the left side of the equation. This will create a perfect square trinomial. ${x^2 + 12x + 36 = -25 + 36}$

3. Rewrite as a squared binomial: The left side is now a perfect square trinomial, which can be written as ${(x + 6)^2}$. ${(x + 6)^2 = 11}$

So, the equation in the required form is: ${(x + 6)^2 = 11}$

(b) Solving the Equation

Now that we have the equation in the form ${(x + 6)^2 = 11}$, we can solve for ${x}$.

1. Take the square root of both sides: ${\sqrt{(x + 6)^2} = \pm\sqrt{11}}$ ${x + 6 = \pm\sqrt{11}}$

2. Isolate ${x}$: Subtract 6 from both sides: ${x = -6 \pm \sqrt{11}}$

Thus, the solutions are: ${x = -6 + \sqrt{11}}$ ${x = -6 - \sqrt{11}}$

Detailed Explanation of Each Step

Moving the Constant Term

The initial equation is ${x^2 + 12x + 25 = 0}$. To begin completing the square, we isolate the terms containing ${x}$ on one side of the equation. This is done by subtracting the constant term, 25, from both sides. This step sets up the equation for the next phase, where we'll add a value to both sides to create a perfect square trinomial. By moving the constant to the right side, we focus on manipulating the left side to fit the ${(x + a)^2}$ pattern. Subtracting 25 from both sides ensures that the equation remains balanced, maintaining equality throughout the process. This is a crucial preliminary step that simplifies the subsequent steps in completing the square. Now, we have ${x^2 + 12x = -25}$, which is ready for completing the square.

Completing the Square

To complete the square for the expression ${x^2 + 12x}$, we need to add a value that turns it into a perfect square trinomial. This value is determined by taking half of the coefficient of the ${x}$ term and then squaring it. In this case, the coefficient of ${x}$ is 12. Half of 12 is 6, and 6 squared is 36. Thus, we add 36 to both sides of the equation ${x^2 + 12x = -25}$. Adding 36 to the left side completes the square, creating the perfect square trinomial ${x^2 + 12x + 36}$. To maintain the equality of the equation, we must also add 36 to the right side, resulting in ${-25 + 36}$. This step is critical because it transforms the left side into a form that can be easily factored into the square of a binomial. By adding the correct value, we can rewrite the equation in the desired format, ${(x + a)^2 = b}$, which is essential for solving for ${x}$. After completing the square, the equation becomes ${x^2 + 12x + 36 = -25 + 36}$.

Rewriting as a Squared Binomial

After completing the square, the left side of the equation ${x^2 + 12x + 36 = -25 + 36}$ is a perfect square trinomial. This trinomial can be factored into the square of a binomial. Specifically, ${x^2 + 12x + 36}$ can be written as ${(x + 6)^2}$. The right side of the equation simplifies to ${-25 + 36 = 11}$. Therefore, the equation can be rewritten as ${(x + 6)^2 = 11}$. This form is crucial because it allows us to easily solve for ${x}$ by taking the square root of both sides. The transformation into a squared binomial simplifies the equation and sets the stage for isolating ${x}$. This step highlights the power of completing the square in converting a quadratic equation into a more manageable form. Now, the equation is ready for the final steps of solving for ${x}$.

Taking the Square Root

To solve the equation ${(x + 6)^2 = 11}$, we begin by taking the square root of both sides. This operation undoes the square on the left side, allowing us to isolate ${x + 6}$. When taking the square root of a number, it's important to remember that there are two possible solutions: a positive square root and a negative square root. Therefore, taking the square root of both sides gives us ${\sqrt{(x + 6)^2} = \pm\sqrt{11}}$, which simplifies to ${x + 6 = \pm\sqrt{11}}$. The ${\pm}$ symbol indicates that we have two possible values for ${x + 6}$: ${\sqrt{11}}$ and ${-\sqrt{11}}$. This step is essential for finding both solutions to the quadratic equation. By considering both the positive and negative square roots, we ensure that we find all possible values of ${x}$ that satisfy the original equation. Now, we can isolate ${x}$ to find the solutions.

Isolating x

After taking the square root, we have ${x + 6 = \pm\sqrt{11}}$. To isolate ${x}$, we subtract 6 from both sides of the equation. This gives us ${x = -6 \pm \sqrt{11}}$. The ${\pm}$ symbol indicates that we have two solutions: ${x = -6 + \sqrt{11}}$ and ${x = -6 - \sqrt{11}}$. These are the two values of ${x}$ that satisfy the original quadratic equation ${x^2 + 12x + 25 = 0}$. The solutions are irrational numbers because ${\sqrt{11}}$ is an irrational number. This final step completes the process of completing the square and provides the exact solutions to the quadratic equation. Therefore, the solutions are ${x = -6 + \sqrt{11}}$ and ${x = -6 - \sqrt{11}}$.

Final Answer

(a) The appropriate form is: ${(x + 6)^2 = 11}$

(b) The solutions to the equation are: ${x = -6 + \sqrt{11}}$ ${x = -6 - \sqrt{11}}$

So, guys, that’s how you solve the quadratic equation ${x^2 + 12x + 25 = 0}$ by completing the square! Hope this helps, and keep up the great work!