Simplifying (2x^2 - 98) / (x^2 + 9x + 14) A Step-by-Step Guide

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Hey guys! Ever stumbled upon a rational expression that looks like a mathematical monster? You know, those fractions with polynomials in the numerator and denominator? Well, fear not! Today, we're going to break down how to simplify these expressions, turning those mathematical monsters into manageable math kittens. We'll use the specific example of (2x^2 - 98) / (x^2 + 9x + 14) to illustrate the process, but the principles we cover will apply to a wide range of rational expressions. So, let's dive in and make some math magic happen!

1. Factoring: The Key to Simplification

The golden rule of simplifying rational expressions is factoring. Think of it as breaking down the expression into its fundamental building blocks. Just like you can't simplify a fraction like 6/8 without recognizing that 6 = 2 * 3 and 8 = 2 * 4, you can't simplify a rational expression without factoring the numerator and denominator.

In our case, we have the expression (2x^2 - 98) / (x^2 + 9x + 14). Let's start by tackling the numerator, 2x^2 - 98. Notice that both terms have a common factor of 2. We can factor this out:

2x^2 - 98 = 2(x^2 - 49)

Now, look closely at what's inside the parentheses: x^2 - 49. This should ring a bell! It's a classic example of the difference of squares. Remember the formula: a^2 - b^2 = (a + b)(a - b). In our case, a = x and b = 7 (since 49 = 7^2). So, we can factor further:

2(x^2 - 49) = 2(x + 7)(x - 7)

Alright, numerator = conquered! Now, let's move on to the denominator, x^2 + 9x + 14. This is a quadratic expression, and we need to find two numbers that add up to 9 (the coefficient of the x term) and multiply to 14 (the constant term). After a little thought, you'll see that 2 and 7 fit the bill (2 + 7 = 9 and 2 * 7 = 14). So, we can factor the denominator as:

x^2 + 9x + 14 = (x + 2)(x + 7)

See how factoring is like unlocking a secret code? We've transformed our original expression into a form where we can clearly see the building blocks.

2. Identifying and Canceling Common Factors

Now that we've factored both the numerator and denominator, we can rewrite our expression as:

(2(x + 7)(x - 7)) / ((x + 2)(x + 7))

Here's where the magic happens! Look for factors that appear in both the numerator and the denominator. These are the common factors that we can cancel out. In our case, we have the factor (x + 7) in both the numerator and the denominator. We can cancel these out, just like simplifying a regular fraction by dividing both the top and bottom by a common factor.

After canceling the (x + 7) factors, we're left with:

(2(x - 7)) / (x + 2)

We're getting closer to our simplified form! This step is crucial because it eliminates the common factors that were making the expression look more complicated than it actually is.

3. The Simplified Expression and Restrictions

We've done the heavy lifting! Our simplified expression is:

(2(x - 7)) / (x + 2)

We can also distribute the 2 in the numerator to get:

(2x - 14) / (x + 2)

This is the simplified form of our original rational expression. We've successfully tamed the mathematical monster!

But, there's one more crucial piece to the puzzle: restrictions. Remember that we can't divide by zero in mathematics. This means we need to identify any values of x that would make the denominator of our original expression equal to zero. These values are not allowed, and we need to state them as restrictions.

Looking back at our factored denominator, (x + 2)(x + 7), we see that the denominator would be zero if x = -2 or x = -7. Therefore, our restrictions are:

x ≠ -2 and x ≠ -7

It's super important to state these restrictions because the simplified expression is only equivalent to the original expression for values of x that are not -2 or -7. If we plug in x = -2 or x = -7 into the simplified expression, we'll get a defined value, but plugging it into the original expression would result in division by zero, which is undefined.

4. Putting it All Together: The Complete Solution

So, to summarize, here's the complete solution for simplifying the rational expression (2x^2 - 98) / (x^2 + 9x + 14):

  • Simplified Expression: (2x - 14) / (x + 2)
  • Restrictions: x ≠ -2 and x ≠ -7

We've successfully simplified the expression and identified the values that x cannot be. That's a math victory!

Let's recap the key concepts we've covered in this guide. Understanding these principles will empower you to simplify a wide range of rational expressions:

  • Factoring is King: The most important step in simplifying rational expressions is factoring the numerator and denominator. This allows you to identify common factors.
  • Common Factors Must Go: Cancel out any factors that appear in both the numerator and the denominator.
  • Simplified Form: After canceling common factors, you're left with the simplified expression.
  • Restrictions are Essential: Identify any values of the variable that would make the original denominator equal to zero. These are the restrictions, and they must be stated as part of the complete solution.
  • Difference of Squares: Remember the formula a^2 - b^2 = (a + b)(a - b). This is a common pattern that appears in factoring problems.
  • Quadratic Factoring: Practice factoring quadratic expressions (ax^2 + bx + c). Look for two numbers that add up to b and multiply to c.

Now that you've got the theory down, it's time to put your knowledge into practice! The best way to master simplifying rational expressions is to work through examples and exercises. Let's look at a few more examples and then give you some exercises to try on your own.

Example 1:

Simplify (x^2 - 4) / (x^2 + 4x + 4)

  1. Factor:
    • Numerator: x^2 - 4 = (x + 2)(x - 2) (Difference of Squares)
    • Denominator: x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2
  2. Rewrite:
    • ((x + 2)(x - 2)) / ((x + 2)(x + 2))
  3. Cancel Common Factors:
    • Cancel one (x + 2) factor from the numerator and denominator.
  4. Simplified Expression:
    • (x - 2) / (x + 2)
  5. Restrictions:
    • x ≠ -2 (because the original denominator, (x + 2)^2, would be zero when x = -2)

Example 2:

Simplify (3x^2 + 9x) / (x^2 + 6x + 9)

  1. Factor:
    • Numerator: 3x^2 + 9x = 3x(x + 3)
    • Denominator: x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2
  2. Rewrite:
    • (3x(x + 3)) / ((x + 3)(x + 3))
  3. Cancel Common Factors:
    • Cancel one (x + 3) factor from the numerator and denominator.
  4. Simplified Expression:
    • (3x) / (x + 3)
  5. Restrictions:
    • x ≠ -3 (because the original denominator, (x + 3)^2, would be zero when x = -3)

Exercises for You to Try:

Okay, your turn! Try simplifying these rational expressions on your own. Remember to factor, cancel common factors, and state the restrictions.

  1. (x^2 - 16) / (x^2 + 8x + 16)
  2. (2x^2 - 8) / (x^2 - 4x + 4)
  3. (x^2 + 5x + 6) / (x^2 + 2x)

Simplifying rational expressions might seem daunting at first, but with a solid understanding of factoring and a little practice, you'll become a pro in no time! Remember the key steps: factor, cancel, simplify, and state restrictions. By mastering these techniques, you'll be able to tackle even the most complex rational expressions with confidence. So go forth, simplify, and conquer those mathematical monsters!

If you're still feeling a little unsure, don't worry! There are tons of resources available online, including videos, practice problems, and detailed explanations. Keep practicing, and you'll get there! Math is a journey, not a destination. Enjoy the ride!