intercepts of f(x)=(x-4)/(x^2-12x+32)

On the horizontal axis, y = 0: solve

\frac{x - 4}{x^{2} - 12 x + 32} = 0

1) Let's solve in

ℝ

\frac{x - 4}{x^{2} - 12 x + 32} = 0

A quotient is zero if and only if its numerator is zero and its denominator is not: we solve the numerator equal to 0

⇥1.1) Let's solve in

ℝ

x - 4 = 0

⇥Candidates, pending the domain check:

x = 4

⇥1.2) To prevent the denominator from being zero, let's solve:

x^{2} - 12 x + 32 \neq 0

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -12

and

c = 32

⇥

Δ = {(-12)}^{2} - 4 \cdot 1 \cdot 32

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

{(-12)}^{2}

(-12) \cdot (-12)

144

⇥

Δ = 144 - 4 \cdot 32

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 32

4 \cdot (30 + 2)

4 \cdot 30 + 4 \cdot 2

120 + 8

128

⇥

Δ = 144 - 128

⇥ⓘ Start from

144

and move back

128

on the number line:

144 - 128

16

⇥

Δ = 16

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-12) - \sqrt{16}}{2 \cdot 1}

⇥

x_{1} = \frac{12 - \sqrt{16}}{2}

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

\sqrt{16}

4

;

4^{2}

16

⇥

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

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

12 - 4

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{- (-12) + \sqrt{16}}{2 \cdot 1}

⇥

x_{2} = \frac{12 + \sqrt{16}}{2}

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

\sqrt{16}

4

;

4^{2}

16

⇥

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

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

12 + 4

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

⇥Solutions found:

x \notin {4, 8}

For the expression

\frac{x - 4}{x^{2} - 12 x + 32}

to be defined, we must have:

x^{2} - 12 x + 32 \neq 0

x^{2} - 12 x + 32 \neq 0

has already been solved:

x \notin {4, 8}

Domain constraints:

x \in ℝ \ {4, 8}

The graph does not cross the horizontal axis

On the vertical axis, x = 0: calculate f(0)

⇥

\frac{0 - 4}{0^{2} - 12 \cdot 0 + 32}

⇥ⓘ Adding 0 changes nothing:

0 - 4

-4

0 + 32

32

■A product with a zero factor is zero:

-12 \cdot 0

0

⇥

- \frac{4}{0 + 32}

⇥ⓘ Adding 0 changes nothing:

0 + 32

32

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

4

, their greatest common divisor:

- \frac{4}{32}

- \frac{1 \cdot 4}{8 \cdot 4}

- \frac{1}{8}

⇥

- \frac{1}{8}

▶Axis intercepts:

(0, - \frac{1}{8})