Solving $(x-12)(x+4)=9$ By Completing The Square A Step-by-Step Guide

Hey everyone! Today, we're diving into a cool technique in algebra called "completing the square." It's a powerful method for solving quadratic equations, and we're going to use it to tackle the equation $(x-12)(x+4)=9$. So, buckle up and let's get started!

Understanding Completing the Square

Completing the square is a method used to rewrite a quadratic equation in a form that allows us to easily solve for the variable. Quadratic equations, those with an $x^2$ term, can be tricky to solve directly, but completing the square transforms them into a more manageable format. This method is especially handy when the quadratic equation doesn't factor easily.

The general idea behind completing the square is to manipulate the quadratic expression to create a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, like $(x + a)^2$ or $(x - a)^2$. Once we have a perfect square trinomial, we can use the square root property to solve for $x$.

Why Completing the Square Matters

You might be wondering, "Why should I bother learning this method when I already know other ways to solve quadratic equations?" Well, completing the square is not just another tool in your math toolbox; it's a fundamental concept that provides a deeper understanding of quadratic equations. It's the backbone behind the quadratic formula, which you've probably encountered before. Understanding completing the square helps you appreciate where the quadratic formula comes from and how it works.

Moreover, completing the square has applications beyond just solving equations. It's used in various areas of mathematics and physics, such as finding the vertex of a parabola, putting equations of circles and ellipses in standard form, and even in some calculus problems. So, mastering this technique is a worthwhile investment in your mathematical journey.

The Core Idea: Creating the Perfect Square

At its heart, completing the square is about turning a quadratic expression of the form $ax^2 + bx + c$ into the form $a(x + h)^2 + k$. The $(x + h)^2$ part is the perfect square trinomial we're aiming for. To achieve this, we focus on manipulating the original expression by adding and subtracting a specific constant. This constant is carefully chosen to create the perfect square.

Think of it like this: we're taking a puzzle with missing pieces and figuring out exactly what pieces we need to add to complete the picture. In the case of completing the square, the "picture" is the perfect square trinomial, and the "missing piece" is the constant term that makes it a perfect square. Once we have this perfect square, solving the equation becomes much simpler.

Step-by-Step Solution for $(x-12)(x+4)=9$

Okay, let's dive into solving our specific equation: $(x-12)(x+4)=9$. We'll break down each step so you can follow along easily.

Step 1: Expand the Left Side

The first thing we need to do is get rid of those parentheses by expanding the left side of the equation. We use the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last) to multiply the two binomials:

$(x - 12)(x + 4) = x(x) + x(4) - 12(x) - 12(4) = x^2 + 4x - 12x - 48$

Now, let's simplify by combining like terms:

$x^2 + 4x - 12x - 48 = x^2 - 8x - 48$

So, our equation now looks like this:

$x^2 - 8x - 48 = 9$

Step 2: Move the Constant Term to the Right Side

To start the process of completing the square, we need to isolate the $x^2$ and $x$ terms on one side of the equation. We do this by adding 48 to both sides:

$x^2 - 8x - 48 + 48 = 9 + 48$

This simplifies to:

$x^2 - 8x = 57$

Step 3: Complete the Square

This is the heart of the method! To complete the square, we need to add a specific constant to both sides of the equation. This constant will turn the left side into a perfect square trinomial.

Here's how we find that constant:

1. Take the coefficient of the $x$ term (which is -8 in our case).
2. Divide it by 2: $-8 / 2 = -4$
3. Square the result: $(-4)^2 = 16$

So, the constant we need to add is 16. Let's add it to both sides of the equation:

$x^2 - 8x + 16 = 57 + 16$

This simplifies to:

$x^2 - 8x + 16 = 73$

Step 4: Factor the Left Side

The left side of the equation is now a perfect square trinomial. It can be factored into the square of a binomial. Remember, we found that $-8/2 = -4$, so the binomial will be $(x - 4)$:

$x^2 - 8x + 16 = (x - 4)(x - 4) = (x - 4)^2$

Our equation now looks like this:

$(x - 4)^2 = 73$

Step 5: Apply the Square Root Property

Now, we can use the square root property to get rid of the square. This property states that if $a^2 = b$, then $a = \pm\sqrt{b}$. Taking the square root of both sides of our equation, we get:

$\sqrt{(x - 4)^2} = \pm\sqrt{73}$

This simplifies to:

$x - 4 = \pm\sqrt{73}$

Step 6: Solve for $x$

Finally, we isolate $x$ by adding 4 to both sides:

$x = 4 \pm \sqrt{73}$

So, the solutions for $x$ are $4 + \sqrt{73}$ and $4 - \sqrt{73}$.

Final Answer and Conclusion

The solutions to the equation $(x-12)(x+4)=9$ are $x = 4 + \sqrt{73}$ and $x = 4 - \sqrt{73}$. This corresponds to option D in the choices provided.

In conclusion, completing the square is a fantastic technique for solving quadratic equations. It might seem a bit involved at first, but with practice, it becomes a smooth and reliable method. Remember the key steps: expand, isolate, complete the square, factor, and apply the square root property. Keep practicing, and you'll master this skill in no time!