Factoring: Solve Quadratic Equation X^2 - 6x - 92 = -7x - 2

Hey guys! Today, we're diving deep into the world of quadratic equations and tackling a specific problem using the method of factoring. Factoring is a crucial skill in algebra, and mastering it will help you solve various mathematical problems. We'll break down each step, making it super easy to follow along. Our mission today is to solve the quadratic equation x^2 - 6x - 92 = -7x - 2 by factoring. Buckle up, let's get started!

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. These equations pop up in many areas of math and science, so knowing how to solve them is super important.

When we talk about solving a quadratic equation, we mean finding the values of x that make the equation true. These values are also called roots or solutions of the equation. One common method for finding these solutions is factoring. Factoring involves breaking down the quadratic expression into a product of two binomials. Once we've done that, we can use the zero-product property to find the values of x. Understanding these basics will make the entire process much smoother, trust me!

Step 1: Rearrange the Equation

Our original equation is x^2 - 6x - 92 = -7x - 2. The first thing we need to do when solving by factoring is to set the equation equal to zero. This means we need to move all the terms to one side of the equation. To do this, we'll add 7x and add 2 to both sides of the equation. Adding the same terms to both sides keeps the equation balanced, just like a scale! So, let's do it:

x^2 - 6x - 92 + 7x + 2 = -7x - 2 + 7x + 2

Now, let's simplify by combining like terms. On the left side, we have -6x and +7x, which combine to give us +x. We also have -92 and +2, which combine to give us -90. On the right side, -7x + 7x cancels out, and -2 + 2 cancels out, leaving us with zero. Our simplified equation now looks like this:

x^2 + x - 90 = 0

Awesome! We've successfully rearranged the equation into the standard quadratic form. This is a crucial step because it sets us up perfectly for factoring. Without this rearrangement, trying to factor would be like trying to assemble a puzzle without all the pieces. Now, we're ready for the next step, which is the heart of the factoring process.

Step 2: Factoring the Quadratic Expression

Now that we have the equation in the standard form x^2 + x - 90 = 0, we can move on to factoring the quadratic expression. Factoring involves finding two binomials that, when multiplied together, give us the quadratic expression. Think of it like reverse multiplication! To factor x^2 + x - 90, we need to find two numbers that multiply to -90 (the constant term) and add up to 1 (the coefficient of the x term). This might sound tricky, but with a bit of practice, it becomes second nature.

Let's think about the factors of -90. We need one positive and one negative factor since their product is negative. Some possible pairs of factors are: (1 and -90), (-1 and 90), (2 and -45), (-2 and 45), (3 and -30), (-3 and 30), (5 and -18), (-5 and 18), (6 and -15), (-6 and 15), (9 and -10), and (-9 and 10). Remember, we're looking for a pair that adds up to 1.

Looking at our list, the pair that works is -9 and 10 because -9 * 10 = -90 and -9 + 10 = 1. Bingo! Now we can rewrite our quadratic expression in factored form using these numbers. We write it as a product of two binomials:

(x - 9)(x + 10) = 0

This means that the quadratic expression x^2 + x - 90 is equivalent to (x - 9)(x + 10). If you were to multiply out (x - 9)(x + 10), you'd get back to x^2 + x - 90. Factoring is like finding the building blocks of an expression, and we've just found them! Now that we've factored the equation, we're ready to use the zero-product property to find our solutions.

Step 3: Applying the Zero-Product Property

The zero-product property is a fundamental concept in algebra that states if the product of two factors is zero, then at least one of the factors must be zero. In simpler terms, if A * B = 0, then either A = 0 or B = 0 (or both). This property is incredibly useful for solving factored quadratic equations because it allows us to break the problem into simpler equations. We've factored our equation into (x - 9)(x + 10) = 0, so now we can apply the zero-product property.

According to the zero-product property, either (x - 9) = 0 or (x + 10) = 0. This gives us two separate linear equations to solve. Let's take each one and solve for x.

For the first equation, x - 9 = 0, we need to isolate x. To do this, we add 9 to both sides of the equation:

x - 9 + 9 = 0 + 9

This simplifies to:

x = 9

Great! We've found our first solution. Now, let's solve the second equation, x + 10 = 0. Again, we need to isolate x. This time, we subtract 10 from both sides of the equation:

x + 10 - 10 = 0 - 10

This simplifies to:

x = -10

Awesome! We've found our second solution. By applying the zero-product property, we've successfully turned our factored quadratic equation into two simple linear equations and solved them. Now, we have both solutions for our original equation. All that's left is to state our final answer.

Step 4: State the Solutions

We've done the hard work of rearranging, factoring, and applying the zero-product property. Now, it's time to state our solutions clearly. We found that x = 9 and x = -10 are the values that make our original equation, x^2 - 6x - 92 = -7x - 2, true. These are the roots or solutions of the quadratic equation. You can even plug these values back into the original equation to check that they work – it's always a good idea to verify your solutions!

So, our final answer is:

x = 9, -10

We often write the solutions as a set, like this: {9, -10}. This notation simply means that the solutions to the equation are 9 and -10. We've successfully solved the quadratic equation by factoring! Give yourself a pat on the back. Understanding how to solve quadratic equations by factoring is a huge step in mastering algebra, and you've just conquered it.

Conclusion

Solving quadratic equations by factoring might seem daunting at first, but as we've seen, it's a straightforward process when you break it down into steps. First, we rearranged the equation to the standard form. Then, we factored the quadratic expression by finding two numbers that multiply to the constant term and add up to the coefficient of the x term. Next, we applied the zero-product property to split the factored equation into two simpler equations. Finally, we solved those equations to find our solutions.

Remember, practice makes perfect! The more you practice factoring quadratic equations, the quicker and more confident you'll become. Keep an eye out for more mathematical adventures, and never stop exploring the amazing world of algebra. You've got this!