intercepts of f(x)=x^2+2x-3 — step-by-step solution

On the horizontal axis, y = 0: solve

x^{2} + 2 x - 3 = 0

Let's solve in

ℝ

x^{2} + 2 x - 3 = 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 = 2

and

c = -3

Δ = 2^{2} - 4 \cdot 1 \cdot (-3)

ⓘ 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 (-3)

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

- 4 \cdot (-3)

12

Δ = 4 + 12

ⓘ Start from

4

and move forward

12

on the number line:

4 + 12

16

Δ = 16

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

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

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

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

- 2 - 4

-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{16}}{2 \cdot 1}

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

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

- 2 + 4

2

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

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

2

, their greatest common divisor:

\frac{2}{2}

1

x_{2} = 1

Solutions obtained:

x = {-3, 1}

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

⇥

0^{2} + 2 \cdot 0 - 3

⇥ⓘ A product with a zero factor is zero:

2 \cdot 0

0

■Adding 0 changes nothing:

0 - 3

-3

⇥

0 - 3

⇥ⓘ Adding 0 changes nothing:

0 - 3

-3

⇥

-3

▶Axis intercepts:

{(-3, 0), (1, 0), (0, -3)}