intercepts of f(x)=-4x^2-8x-3 — step-by-step solution

On the horizontal axis, y = 0: solve

- 4 x^{2} - 8 x - 3 = 0

Let's solve in

ℝ

- 4 x^{2} - 8 x - 3 = 0

Since

-1 \neq 0

, divide both sides by

-1

4 x^{2} + 8 x + 3 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 4

b = 8

and

c = 3

Δ = 8^{2} - 4 \cdot 4 \cdot 3

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

8^{2}

8 \cdot 8

64

■Multiplications are done before additions and subtractions:

4 \cdot 4

16

Δ = 64 - 16 \cdot 3

ⓘ Multiplications are done before additions and subtractions:

16 \cdot 3

(10 + 6) \cdot 3

10 \cdot 3 + 6 \cdot 3

30 + 18

48

Δ = 64 - 48

ⓘ Start from

64

and move back

48

on the number line:

64 - 48

16

Δ = 16

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 8 - \sqrt{16}}{2 \cdot 4}

x_{1} = \frac{- 8 - \sqrt{16}}{2 \cdot 4}

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

\sqrt{16}

4

;

4^{2}

16

ⅰ

2 \cdot 4

8

x_{1} = \frac{- 8 - 4}{8}

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

- 8 - 4

-12

x_{1} = - \frac{12}{8}

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

4

, their greatest common divisor:

- \frac{12}{8}

- \frac{3 \cdot 4}{2 \cdot 4}

- \frac{3}{2}

x_{1} = - \frac{3}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 8 + \sqrt{16}}{2 \cdot 4}

x_{2} = \frac{- 8 + \sqrt{16}}{2 \cdot 4}

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

\sqrt{16}

4

;

4^{2}

16

ⅰ

2 \cdot 4

8

x_{2} = \frac{- 8 + 4}{8}

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

- 8 + 4

-4

x_{2} = - \frac{4}{8}

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

4

, their greatest common divisor:

- \frac{4}{8}

- \frac{1 \cdot 4}{2 \cdot 4}

- \frac{1}{2}

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

Solutions obtained:

x = {- \frac{3}{2}, - \frac{1}{2}}

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

⇥

- 4 \cdot 0^{2} - 8 \cdot 0 - 3

⇥ⓘ A product with a zero factor is zero:

-4 \cdot 0

0

-8 \cdot 0

0

⇥

0 + 0 - 3

⇥ⓘ Adding 0 changes nothing:

0 + 0 - 3

-3

⇥

-3

▶Axis intercepts:

{(- \frac{3}{2}, 0), (- \frac{1}{2}, 0), (0, -3)}