intercepts of f(x)=x^2+x-20 — step-by-step solution

On the horizontal axis, y = 0: solve

x^{2} + x - 20 = 0

Let's solve in

ℝ

x^{2} + x - 20 = 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 = 1

and

c = -20

Δ = 1 - 4 \cdot 1 \cdot (-20)

Δ = 1 - 4 \cdot (-20)

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

- 4 \cdot (-20)

80

Δ = 1 + 80

ⓘ Start from

1

and move forward

80

on the number line:

1 + 80

81

Δ = 81

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 1 - \sqrt{81}}{2 \cdot 1}

x_{1} = \frac{- 1 - \sqrt{81}}{2}

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

\sqrt{81}

9

;

9^{2}

81

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

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

- 1 - 9

-10

x_{1} = - \frac{10}{2}

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

2

, their greatest common divisor:

- \frac{10}{2}

-5

x_{1} = -5

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 1 + \sqrt{81}}{2 \cdot 1}

x_{2} = \frac{- 1 + \sqrt{81}}{2}

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

\sqrt{81}

9

;

9^{2}

81

x_{2} = \frac{- 1 + 9}{2}

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

- 1 + 9

8

x_{2} = \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_{2} = 4

Solutions found:

x = {-5, 4}

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

⇥

0^{2} + 0 - 20

⇥ⓘ Adding 0 changes nothing:

0 + 0 - 20

-20

⇥

-20

▶Axis intercepts:

{(-5, 0), (4, 0), (0, -20)}