Factoring G(x) = 2x² - 6x - 56: A Step-by-Step Solution

Hey everyone! Today, we're diving into the world of quadratic equations, specifically focusing on factoring. We'll be tackling a problem where we need to express a quadratic function in its factored form. This is a crucial skill in algebra, and once you grasp the concept, you'll find it super useful in solving various mathematical problems. So, let's jump right in and break down the equation g(x) = 2x² - 6x - 56.

Understanding Factored Form

Before we get our hands dirty with the equation, let's quickly recap what factored form actually means. A quadratic equation in its standard form looks like this: ax² + bx + c. The factored form, on the other hand, expresses the quadratic as a product of two binomials, like this: a(x - r)(x - s), where 'r' and 's' are the roots or zeros of the equation. Finding the factored form helps us easily identify these roots, which are the values of x that make the equation equal to zero.

So, when we talk about putting g(x) = 2x² - 6x - 56 into factored form, we're essentially trying to rewrite it as a product of simpler expressions. This involves a few key steps, which we'll walk through together.

Step 1: Factor out the Greatest Common Factor (GCF)

Our equation is g(x) = 2x² - 6x - 56. The first thing we should always look for is a common factor among all the terms. In this case, we can see that 2 is a common factor for 2x², -6x, and -56. Factoring out the 2 simplifies our equation and makes it easier to work with. So, let's do that:

g(x) = 2(x² - 3x - 28)

Now, we have a simpler quadratic expression inside the parentheses: x² - 3x - 28. This is much more manageable, right? Factoring out the GCF is like decluttering before you start a big project – it clears the way and makes the rest of the process smoother.

Step 2: Factoring the Quadratic Expression

Now, let's focus on the expression inside the parentheses: x² - 3x - 28. To factor this, we need to find two numbers that multiply to give us -28 (the constant term) and add up to -3 (the coefficient of the x term). This might sound like a puzzle, but with a bit of practice, you'll become a pro at spotting these numbers.

Think about the factors of -28. We have pairs like 1 and -28, -1 and 28, 2 and -14, -2 and 14, 4 and -7, and -4 and 7. Among these pairs, which one adds up to -3? Bingo! It's 4 and -7. So, we can rewrite our quadratic expression as:

x² - 3x - 28 = (x + 4)(x - 7)

This is the heart of factoring. We've broken down the quadratic into two binomial factors. Remember, the order matters here. (x + 4) and (x - 7) are the specific factors that give us the original quadratic when multiplied together. To double-check, you can always use the FOIL method (First, Outer, Inner, Last) to expand the factored form and see if it matches the original expression.

Step 3: Write the Complete Factored Form

Now that we've factored the quadratic expression inside the parentheses, we need to remember the 2 we factored out in the first step. We simply put it back in front of our factored expression:

g(x) = 2(x + 4)(x - 7)

And there you have it! This is the factored form of our original equation, g(x) = 2x² - 6x - 56. We've successfully transformed the equation from its standard form to its factored form, making it easier to analyze and solve.

Analyzing the Answer Choices

Now, let's take a look at the answer choices provided and see which one matches our factored form:

A. g(x) = 2(x + 4)(x - 7) B. g(x) = 2(x² - 3x - 28) C. g(x) = 2(x - 4)(x + 7) D. g(x) = (2x + 7)(x - 8)

As you can clearly see, option A, g(x) = 2(x + 4)(x - 7), is the correct answer. It perfectly matches the factored form we derived step-by-step. Option B is the expression after taking out the GCF but not fully factored. Options C and D have incorrect factors.

Choosing the correct answer becomes straightforward when you understand the process of factoring and can confidently apply it to the given equation. It’s not just about picking the right option; it’s about knowing why it’s the right option.

Why Factored Form Matters

You might be wondering,