Factor Quadratic Equations: Step-by-Step Solution

by Viktoria Ivanova 50 views

Hey guys! Let's dive into the world of quadratic functions and explore a super useful technique for solving them: factoring. Trust me, once you get the hang of this, you'll be able to tackle these problems like a pro. In this guide, we'll break down the process step-by-step, using a specific example to make it crystal clear. So, grab your pencils, and let's get started!

Understanding Quadratic Functions

Before we jump into factoring, let's quickly recap what a quadratic function actually is. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (usually x) is 2. The general form of a quadratic function is:

f(x) = ax² + bx + c

Where a, b, and c are constants, and a is not equal to zero (if a were zero, it would be a linear function, not a quadratic!). The graph of a quadratic function is a parabola, a U-shaped curve that can open upwards or downwards depending on the sign of a.

Solving a quadratic function means finding the values of x for which f(x) = 0. These values are also known as the roots, zeros, or x-intercepts of the function. There are several ways to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Factoring is often the quickest and easiest method when it's applicable.

In our example, we have the quadratic function:

y = x² - 2x - 48

Our goal is to find the values of x that make y equal to zero. This is where factoring comes in handy.

The Power of Factoring Quadratic Equations

Factoring is like reverse multiplication. When we factor a quadratic expression, we're essentially trying to find two binomials that, when multiplied together, give us the original quadratic. This method is based on the principle that if the product of two factors is zero, then at least one of the factors must be zero. This is a fundamental concept in algebra and is key to solving quadratic equations by factoring.

Think of it this way: if we can rewrite our quadratic equation in the form:

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

Then, either (x + p) = 0 or (x + q) = 0. Solving these two simple linear equations will give us the solutions (roots) of the quadratic equation. But how do we find those binomial factors in the first place? That's what we'll explore next.

Factoring by Finding the Right Numbers

The core of factoring a quadratic expression like x² - 2x - 48 lies in finding two numbers that satisfy two specific conditions. These numbers, let's call them p and q, need to:

  1. Multiply to the constant term (c), which in our case is -48.
  2. Add up to the coefficient of the x term (b), which in our case is -2.

This might sound a bit like a puzzle, and that's because it is! But don't worry, there's a systematic way to approach this. We start by listing the factors of the constant term (-48) and then see which pair adds up to the coefficient of the x term (-2).

Let's list the factor pairs of -48:

  • 1 and -48
  • -1 and 48
  • 2 and -24
  • -2 and 24
  • 3 and -16
  • -3 and 16
  • 4 and -12
  • -4 and 12
  • 6 and -8
  • -6 and 8

Now, let's examine these pairs to see which one adds up to -2. Looking closely, we can see that the pair 6 and -8 fits the bill:

6 * (-8) = -48 6 + (-8) = -2

Bingo! We've found our magic numbers.

Constructing the Factors

Now that we've identified our numbers (6 and -8), we can construct the factored form of the quadratic expression. Remember, we're aiming for the form (x + p)(x + q). In our case, p is 6 and q is -8. So, we can write the factored form as:

(x + 6)(x - 8)

Notice that we've replaced q with -8, making it a subtraction within the second binomial. This is crucial for getting the signs right when multiplying the factors back out.

Verification Through Expansion

To be absolutely sure we've factored correctly, it's always a good idea to expand the factored form and see if it matches the original quadratic expression. We can do this using the FOIL method (First, Outer, Inner, Last):

(x + 6)(x - 8)

  • First: x * x = x²
  • Outer: x * -8 = -8x
  • Inner: 6 * x = 6x
  • Last: 6 * -8 = -48

Now, let's combine the terms:

x² - 8x + 6x - 48 = x² - 2x - 48

Success! Our expanded form matches the original quadratic expression, y = x² - 2x - 48, confirming that our factoring is correct.

Finding the Solutions

We're not quite done yet! Remember, our ultimate goal is to solve the quadratic equation, meaning find the values of x that make y equal to zero. Now that we have the factored form, this becomes a simple task. We use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

So, we have:

(x + 6)(x - 8) = 0

This means either:

x + 6 = 0 or x - 8 = 0

Let's solve these two equations separately:

  1. x + 6 = 0 Subtract 6 from both sides: x = -6

  2. x - 8 = 0 Add 8 to both sides: x = 8

Therefore, the solutions to the quadratic equation y = x² - 2x - 48 are x = -6 and x = 8. These are the x-intercepts of the parabola represented by this equation.

Enclosing the Number

Now, let's go back to the original question. We were asked to factor the quadratic and enclose the number that belongs in the green box. The factored form we found was:

y = (x + 6)(x - 8)

Based on this, the number that belongs in the green box (corresponding to the p value in (x + p)) is 6. Woohoo! We did it!

Conclusion: Mastering Quadratic Factoring

Factoring quadratic functions might seem a little tricky at first, but with practice, it becomes a powerful tool in your mathematical arsenal. By understanding the relationship between the factors and the coefficients of the quadratic expression, you can efficiently find the roots of the equation. Remember the key steps:

  1. Identify the a, b, and c coefficients.
  2. Find two numbers that multiply to c and add up to b.
  3. Construct the factored form using these numbers.
  4. Verify your factoring by expanding.
  5. Set each factor equal to zero and solve for x.

So, next time you encounter a quadratic equation, don't panic! Take a deep breath, remember the factoring process, and you'll be solving them in no time. Keep practicing, and you'll become a quadratic-solving master! Keep an eye out for more math guides and tips. Happy factoring, guys! Remember that math is a journey, and every problem solved is a step forward. Stay curious, keep learning, and don't be afraid to ask questions. You've got this!