intercepts of -4x^2+6x-1 — step-by-step solution

On the horizontal axis, y = 0: solve

- 4 x^{2} + 6 x - 1 = 0

Let's solve in

ℝ

- 4 x^{2} + 6 x - 1 = 0

Since

-1 \neq 0

, divide both sides by

-1

4 x^{2} - 6 x + 1 = 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 = -6

and

c = 1

Δ = {(-6)}^{2} - 4 \cdot 4 \cdot 1

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

{(-6)}^{2}

(-6) \cdot (-6)

36

Δ = 36 - 4 \cdot 4

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 4

16

Δ = 36 - 16

ⓘ Start from

36

and move back

16

on the number line:

36 - 16

20

Δ = 20

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-6) - \sqrt{20}}{2 \cdot 4}

x_{1} = \frac{6 - \sqrt{20}}{2 \cdot 4}

ⓘ We look for the largest square dividing

20

4

2^{2}

, and take it out of the root:

\sqrt{20}

\sqrt{4 \cdot 5}

\sqrt{4} \cdot \sqrt{5}

2 \cdot \sqrt{5}

ⅰ

2 \cdot 4

8

x_{1} = \frac{6 - 2 \cdot \sqrt{5}}{8}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{6 - 2 \cdot \sqrt{5}}{8}

\frac{3 - \sqrt{5}}{4}

x_{1} = \frac{3 - \sqrt{5}}{4}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-6) + \sqrt{20}}{2 \cdot 4}

x_{2} = \frac{6 + \sqrt{20}}{2 \cdot 4}

ⓘ We look for the largest square dividing

20

4

2^{2}

, and take it out of the root:

\sqrt{20}

\sqrt{4 \cdot 5}

\sqrt{4} \cdot \sqrt{5}

2 \cdot \sqrt{5}

ⅰ

2 \cdot 4

8

x_{2} = \frac{6 + 2 \cdot \sqrt{5}}{8}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{6 + 2 \cdot \sqrt{5}}{8}

\frac{3 + \sqrt{5}}{4}

x_{2} = \frac{3 + \sqrt{5}}{4}

Solutions found:

x = {\frac{3 - \sqrt{5}}{4}, \frac{3 + \sqrt{5}}{4}}

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

⇥

- 4 \cdot 0^{2} + 6 \cdot 0 - 1

⇥ⓘ A product with a zero factor is zero:

-4 \cdot 0

0

6 \cdot 0

0

⇥

0 + 0 - 1

⇥ⓘ Adding 0 changes nothing:

0 + 0 - 1

-1

⇥

-1

▶Axis intercepts:

{(\frac{3 - \sqrt{5}}{4}, 0), (\frac{3 + \sqrt{5}}{4}, 0), (0, -1)}