Solve X By Factoring: X^2 - X - 35 = -x + 1

Introduction

Hey guys! Today, we're diving into a classic algebra problem: solving for the values of x in a quadratic equation by factoring. Factoring is a super useful technique in algebra, and mastering it can make solving equations like this one a breeze. We'll take a step-by-step approach to break down the problem, making sure you understand each part of the process. So, let's get started and tackle this equation together!

Problem Statement: x^2 - x - 35 = -x + 1

The equation we need to solve is x^2 - x - 35 = -x + 1. Our mission is to find all possible values of x that make this equation true. Factoring is a powerful method for solving quadratic equations, especially when the equation can be neatly expressed as a product of two binomials. Before we jump into factoring, we need to rearrange the equation into the standard quadratic form, which is ax^2 + bx + c = 0. This form helps us identify the coefficients and constants that we'll use in the factoring process. So, let's get this equation into the right shape and then start factoring!

Step 1: Rearrange the Equation

The first thing we need to do is get our equation into that standard quadratic form: ax^2 + bx + c = 0. To do this, we'll need to move all the terms to one side of the equation, leaving zero on the other side. We start with the original equation: x^2 - x - 35 = -x + 1. To eliminate the -x on the right side, we'll add x to both sides of the equation. This gives us: x^2 - x + x - 35 = -x + x + 1, which simplifies to x^2 - 35 = 1. Next, we need to get rid of the +1 on the right side. We do this by subtracting 1 from both sides: x^2 - 35 - 1 = 1 - 1. This simplifies to our standard form: x^2 - 36 = 0. Now that we have the equation in this form, we can easily see that a = 1, b = 0 (since there's no x term), and c = -36. This sets us up perfectly for factoring!

Step 2: Factoring the Quadratic Expression

Now that we've got our equation in the standard form x^2 - 36 = 0, it's time to factor. When we look at x^2 - 36, we should recognize a special pattern: the difference of squares. This pattern is in the form a^2 - b^2, which can be factored into (a - b)(a + b). In our case, x^2 is our a^2 and 36 is our b^2. So, we need to figure out what a and b are. The square root of x^2 is x, so a = x. The square root of 36 is 6, so b = 6. Applying the difference of squares pattern, we can factor x^2 - 36 into (x - 6)(x + 6). So, our equation x^2 - 36 = 0 becomes (x - 6)(x + 6) = 0. This factored form is super helpful because it breaks down the quadratic expression into two simpler binomials. Now, we can use the zero-product property to find our solutions.

Step 3: Apply the Zero-Product Property

We've successfully factored our equation into (x - 6)(x + 6) = 0. Now comes the really cool part: the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In simpler terms, if we have two things multiplied together that equal zero, then either the first thing is zero, the second thing is zero, or both are zero. Applying this to our factored equation, it means that either (x - 6) = 0 or (x + 6) = 0. This gives us two separate, simpler equations to solve. For the first equation, x - 6 = 0, we just need to add 6 to both sides to isolate x. This gives us x = 6. For the second equation, x + 6 = 0, we subtract 6 from both sides to isolate x. This gives us x = -6. So, we have two potential solutions: x = 6 and x = -6. Let's check if these solutions actually work in our original equation.

Step 4: Check the Solutions

We've found two potential solutions for x: x = 6 and x = -6. It’s always a good idea to check our solutions to make sure they’re correct and that we haven’t made any mistakes along the way. To do this, we’ll plug each value of x back into our original equation, x^2 - x - 35 = -x + 1, and see if both sides of the equation are equal. First, let’s check x = 6: Substituting x = 6 into the equation gives us (6)^2 - 6 - 35 = -6 + 1. Simplifying the left side, we get 36 - 6 - 35 = -5, which further simplifies to -5 = -5. On the right side, we have -6 + 1 = -5. Since both sides are equal, x = 6 is indeed a solution. Now, let’s check x = -6: Substituting x = -6 into the equation gives us (-6)^2 - (-6) - 35 = -(-6) + 1. Simplifying the left side, we get 36 + 6 - 35 = 7, which further simplifies to 7 = 7. On the right side, we have 6 + 1 = 7. Since both sides are equal, x = -6 is also a solution. Great! Both solutions check out.

Final Answer

After rearranging the equation, factoring, applying the zero-product property, and checking our solutions, we've successfully solved for x. The values of x that satisfy the equation x^2 - x - 35 = -x + 1 are x = 6 and x = -6. These are the two numbers that, when plugged back into the original equation, make the equation true. Factoring can be a really powerful tool when you're solving quadratic equations, and this problem is a great example of how it works. Keep practicing, and you'll become a factoring pro in no time!