intercepts of f(x)=x^2+22x+117 — step-by-step solution

On the horizontal axis, y = 0: solve

x^{2} + 22 x + 117 = 0

Let's solve in

ℝ

x^{2} + 22 x + 117 = 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 = 22

and

c = 117

Δ = 22^{2} - 4 \cdot 1 \cdot 117

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

22^{2}

22 \cdot 22

484

Δ = 484 - 4 \cdot 117

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 117

4 \cdot (100 + 10 + 7)

4 \cdot 100 + 4 \cdot 10 + 4 \cdot 7

400 + 40 + 28

468

Δ = 484 - 468

ⓘ Start from

484

and move back

468

on the number line:

484 - 468

16

Δ = 16

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 22 - \sqrt{16}}{2 \cdot 1}

x_{1} = \frac{- 22 - \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{- 22 - 4}{2}

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

- 22 - 4

-26

x_{1} = - \frac{26}{2}

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

2

, their greatest common divisor:

- \frac{26}{2}

-13

x_{1} = -13

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 22 + \sqrt{16}}{2 \cdot 1}

x_{2} = \frac{- 22 + \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{- 22 + 4}{2}

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

- 22 + 4

-18

x_{2} = - \frac{18}{2}

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

2

, their greatest common divisor:

- \frac{18}{2}

-9

x_{2} = -9

Solutions obtained:

x = {-13, -9}

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

⇥

0^{2} + 22 \cdot 0 + 117

⇥ⓘ A product with a zero factor is zero:

22 \cdot 0

0

■Adding 0 changes nothing:

0 + 117

117

⇥

0 + 117

⇥ⓘ Adding 0 changes nothing:

0 + 117

117

⇥

117

▶Axis intercepts:

{(-13, 0), (-9, 0), (0, 117)}