Algebra

(5x^2 - 20) / (5x^2 - 9x - 2)

Step-by-step solution with explanation

Final Answer

, where and

Step-by-step solution

Factor the numerator completely

First pull out the common factor of 5, then recognize x² - 4 as a difference of squares. A difference of squares factors as (a+b)(a-b).

Factor the denominator completely

We need two factors of the form (5x + a)(x + b) where a·b = -2 and 5b + a = -9. Testing (5x + 1)(x - 2): 5x·x + 5x·(-2) + 1·x + 1·(-2) = 5x² - 10x + x - 2 = 5x² - 9x - 2. ✓

Rewrite the full fraction with factors

Now we can clearly see both the numerator and denominator written in factored form. This sets us up to cancel any shared factors.

Cancel the common factor (x − 2)

The factor (x - 2) appears in both top and bottom, so it cancels. We must note that x ≠ 2 because plugging x = 2 into the original expression causes division by zero.

State the simplified expression and restrictions

The fully simplified form is 5(x+2)/(5x+1). We also exclude x = -1/5 because that makes the denominator 5x+1 equal zero in the simplified expression (and in the original).

Understanding this problem

Learning Insight

Simplifying a rational expression (a fraction with polynomials) works just like simplifying a number fraction — you cancel factors that appear on both top and bottom. The key idea is that (x-2)/(x-2) = 1 for any x ≠ 2, so canceling it doesn't change the value of the expression.

Quick Tip

Always factor COMPLETELY before canceling. You can only cancel entire factors (things multiplied), never terms that are added or subtracted. If you see a difference of squares like a² - b², immediately think (a+b)(a-b).

Common Mistake

Students often cancel individual terms instead of factors — for example, canceling the '5x²' in the numerator with the '5x²' in the denominator before factoring. You can only cancel multiplicative factors, not additive terms.