Factoring Quadratics: Solve X² + 18x = 6x - 35

by Kenji Nakamura 47 views

Hey guys! Today, we're diving deep into the world of quadratic equations, specifically how to solve them by factoring. Factoring might sound intimidating, but trust me, once you get the hang of it, it's like riding a bike (math bike, that is!). We'll break down the process step-by-step, using an example to make things crystal clear. So, grab your pencils and let's get started!

What are Quadratic Equations?

Before we jump into factoring, let's quickly recap what quadratic equations are. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (usually 'x') is 2. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where:

  • 'a', 'b', and 'c' are constants (numbers), and
  • 'x' is the variable.

Think of it like a recipe: 'a', 'b', and 'c' are the ingredients, and 'x' is the mystery flavor we're trying to uncover. Solving a quadratic equation means finding the values of 'x' that make the equation true. These values are also known as the roots or solutions of the equation.

Quadratic equations pop up all over the place in real life, from calculating the trajectory of a baseball to designing bridges. So, mastering them is a pretty valuable skill!

Factoring: The Magic Trick for Solving Quadratics

Okay, now for the main event: factoring. Factoring is like reverse multiplication. Instead of multiplying expressions together, we're breaking down a quadratic expression into the product of two simpler expressions (usually binomials). These binomials are expressions with two terms, like (x + 3) or (2x - 1).

Why do we do this? Because of something called the Zero Product Property. This property is our secret weapon! It states that if the product of two or more factors is zero, then at least one of the factors must be zero. In other words, if we can factor a quadratic equation into the form:

(x + p)(x + q) = 0

Then either (x + p) = 0 or (x + q) = 0. This allows us to solve for 'x' by setting each factor equal to zero and solving the resulting linear equations.

Steps to Solve Quadratic Equations by Factoring

Here's a step-by-step guide to solving quadratic equations by factoring:

  1. Set the Equation to Zero: The first step is crucial! Make sure your quadratic equation is in the standard form (ax² + bx + c = 0). This means getting all the terms on one side of the equation and setting it equal to zero. If your equation isn't already in this form, you'll need to rearrange it by adding or subtracting terms from both sides.
  2. Factor the Quadratic Expression: This is the heart of the factoring process. We need to find two binomials that, when multiplied together, give us the original quadratic expression. There are different techniques for factoring, and we'll explore one in detail in our example. The key is to find the right combination of numbers that satisfy certain conditions related to the coefficients 'a', 'b', and 'c'.
  3. Apply the Zero Product Property: Once you've factored the quadratic expression, you'll have something in the form (x + p)(x + q) = 0. Now, we use the Zero Product Property. Set each factor equal to zero, creating two separate linear equations:
    • x + p = 0
    • x + q = 0
  4. Solve for 'x': Solve each of the linear equations you created in the previous step. This will give you the two solutions (or roots) of the quadratic equation. These are the values of 'x' that make the original equation true.
  5. Check Your Solutions (Optional but Recommended): To be extra sure you've got the right answers, you can plug each solution back into the original quadratic equation. If both sides of the equation are equal, then your solution is correct. This is a great way to catch any mistakes you might have made along the way.

Example Time: Cracking the Code

Let's tackle an example to see these steps in action. We'll use the equation you provided:

x² + 18x = 6x - 35

This equation isn't in standard form yet, so our first step is to rearrange it.

Step 1: Set the Equation to Zero

To get the equation into standard form, we need to move all the terms to the left side. Subtract 6x from both sides:

x² + 18x - 6x = -35

Combine the 'x' terms:

x² + 12x = -35

Now, add 35 to both sides:

x² + 12x + 35 = 0

Great! Now we have our quadratic equation in standard form:

x² + 12x + 35 = 0

Step 2: Factor the Quadratic Expression

This is where the factoring magic happens. We need to find two numbers that:

  • Multiply to give us 35 (the 'c' term)
  • Add up to give us 12 (the 'b' term)

Think about the factors of 35: 1 and 35, or 5 and 7. Which pair adds up to 12? You got it – 5 and 7!

So, we can factor the quadratic expression as:

(x + 5)(x + 7) = 0

If you're unsure, you can always multiply the binomials back together using the FOIL method (First, Outer, Inner, Last) to check if you get the original expression.

Step 3: Apply the Zero Product Property

Now, we use the Zero Product Property. We set each factor equal to zero:

  • x + 5 = 0
  • x + 7 = 0

Step 4: Solve for 'x'

Solve each equation for 'x':

  • For x + 5 = 0, subtract 5 from both sides: x = -5
  • For x + 7 = 0, subtract 7 from both sides: x = -7

So, our solutions are x = -5 and x = -7.

Step 5: Check Your Solutions (Optional)

Let's check our solutions to make sure they work. Plug x = -5 into the original equation:

(-5)² + 18(-5) = 6(-5) - 35

25 - 90 = -30 - 35

-65 = -65

It works!

Now, let's check x = -7:

(-7)² + 18(-7) = 6(-7) - 35

49 - 126 = -42 - 35

-77 = -77

It works too! We've successfully solved the quadratic equation.

Tips and Tricks for Factoring

Factoring can sometimes be tricky, but here are a few tips and tricks to help you along the way:

  • Look for a Greatest Common Factor (GCF): Before you start factoring, always check if there's a GCF that you can factor out of all the terms. This can simplify the expression and make factoring easier.
  • **Use the