-2x^2+8x-4=0 — step-by-step solution

1) Let's solve in

ℝ

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

This expression can be factorized into:

2 \cdot (- x^{2} + 4 x - 2) = 0

2) Let's solve:

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

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

Δ = 4^{2} - 4 \cdot (-1) \cdot (-2)

ℹ

4^{2}

16

‖

-4 \cdot (-1)

4

Δ = 16 - 4 \cdot 2

Δ = 16 - 8

Δ = 8

Since

Δ > 0

, the equation has two solutions:

The first solution is:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{- 4 - \sqrt{8}}{2 \cdot (-1)}

ℹ

- \sqrt{8}

- 2 \cdot \sqrt{2}

‖

2 \cdot (-1)

-2

x1 = \frac{- (4 + 2 \cdot \sqrt{2})}{-2}

x1 = \frac{4 + 2 \cdot \sqrt{2}}{2}

x1 = 2 + \sqrt{2}

The second solution is:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{- 4 + \sqrt{8}}{2 \cdot (-1)}

ℹ

\sqrt{8}

2 \cdot \sqrt{2}

‖

2 \cdot (-1)

-2

x2 = \frac{- 4 + 2 \cdot \sqrt{2}}{-2}

x2 = - (\frac{- 4 + 2 \cdot \sqrt{2}}{2})

x2 = \frac{4 - 2 \cdot \sqrt{2}}{2}

x2 = 2 - \sqrt{2}

Solution found:

x = 2 - \sqrt{2}

Solution obtained:

x = \sqrt{2} + 2

▶Solution:

x = {2 - \sqrt{2}, \sqrt{2} + 2}

3) The numeric value of:

x = {2 - \sqrt{2}, \sqrt{2} + 2}

is:

x = {2 - \approx 1.4142, \approx 1.4142 + 2}

▷

x = {\approx 0.5858, \approx 3.4142}