intercepts of x^2-2x-15 — step-by-step solution

On the horizontal axis, y = 0: solve

x^{2} - 2 x - 15 = 0

Let's solve in

ℝ

x^{2} - 2 x - 15 = 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 = -2

and

c = -15

Δ = {(-2)}^{2} - 4 \cdot 1 \cdot (-15)

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

{(-2)}^{2}

(-2) \cdot (-2)

4

Δ = 4 - 4 \cdot (-15)

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

- 4 \cdot (-15)

4 \cdot 15

4 \cdot (10 + 5)

4 \cdot 10 + 4 \cdot 5

40 + 20

60

Δ = 4 + 60

ⓘ Start from

4

and move forward

60

on the number line:

4 + 60

64

Δ = 64

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-2) - \sqrt{64}}{2 \cdot 1}

x_{1} = \frac{2 - \sqrt{64}}{2}

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

\sqrt{64}

8

;

8^{2}

64

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

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

2 - 8

-6

x_{1} = - \frac{6}{2}

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

2

, their greatest common divisor:

- \frac{6}{2}

-3

x_{1} = -3

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-2) + \sqrt{64}}{2 \cdot 1}

x_{2} = \frac{2 + \sqrt{64}}{2}

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

\sqrt{64}

8

;

8^{2}

64

x_{2} = \frac{2 + 8}{2}

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

2 + 8

10

x_{2} = \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_{2} = 5

Solutions obtained:

x = {-3, 5}

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

⇥

0^{2} - 2 \cdot 0 - 15

⇥ⓘ A product with a zero factor is zero:

-2 \cdot 0

0

■Adding 0 changes nothing:

0 - 15

-15

⇥

0 - 15

⇥ⓘ Adding 0 changes nothing:

0 - 15

-15

⇥

-15

▶Axis intercepts:

{(-3, 0), (5, 0), (0, -15)}