domain of f(x)=(4x+2)/(x^2-4x-32)

We save the definition:

f (x) = \frac{4 x + 2}{x^{2} - 4 x - 32}

Let's evaluate the domain of

\frac{4 x + 2}{x^{2} - 4 x - 32}

For the expression

\frac{4 x + 2}{x^{2} - 4 x - 32}

to exist, you must check:

x^{2} - 4 x - 32 \neq 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -4

and

c = -32

Δ = {(-4)}^{2} - 4 \cdot 1 \cdot (-32)

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-4)}^{2}

(-4) \cdot (-4)

16

Δ = 16 - 4 \cdot (-32)

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-32)

4 \cdot 32

4 \cdot (30 + 2)

4 \cdot 30 + 4 \cdot 2

120 + 8

128

Δ = 16 + 128

ⓘ Start from

16

and move forward

128

on the number line:

16 + 128

144

Δ = 144

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-4) - \sqrt{144}}{2 \cdot 1}

x_{1} = \frac{4 - \sqrt{144}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{144}

12

;

12^{2}

144

x_{1} = \frac{4 - 12}{2}

ⓘ The numerator and the denominator are computed before the division:

4 - 12

-8

x_{1} = - \frac{8}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{8}{2}

-4

x_{1} = -4

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-4) + \sqrt{144}}{2 \cdot 1}

x_{2} = \frac{4 + \sqrt{144}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{144}

12

;

12^{2}

144

x_{2} = \frac{4 + 12}{2}

ⓘ The numerator and the denominator are computed before the division:

4 + 12

16

x_{2} = \frac{16}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{16}{2}

8

x_{2} = 8

▶Domain of definition:

x \in ℝ \ {-4, 8}