factor (x^2-9x+20)/(16-x^2) — step-by-step solution

⚠Ⓘ 9x is read as the product 9*x: for a power, write 9^x

1) The original denominators must be nonzero:

x \in ℝ \ {-4, 4}

Let's factor in

ℝ

\frac{x^{2} - 9 x + 20}{16 - x^{2}}

2) Let's factor in

ℝ

x^{2} - 9 x + 20

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

Δ = {(-9)}^{2} - 4 \cdot 1 \cdot 20

Δ = 81 - 4 \cdot 20

Δ = 81 - 80

Δ = 1

Since

Δ > 0

, the equation has two solutions:

For the first solution, we obtain:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{9 - 1}{2}

x1 = \frac{8}{2}

x1 = 4

The second solution is:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{9 + 1}{2}

x2 = \frac{10}{2}

x2 = 5

Factors identified:

(x - 4) \cdot (x - 5)

3) Let's factor in

ℝ

16 - x^{2}

Factors found:

- (x + 4) \cdot (x - 4)

\frac{(x - 4) \cdot (x - 5)}{- (x + 4) \cdot (x - 4)}

- \frac{(x - 4) \cdot (x - 5)}{(x + 4) \cdot (x - 4)}

- \frac{x - 5}{x + 4}

\frac{- x + 5}{x + 4}

▶Factors identified:

\frac{5 - x}{x + 4}