-2x^2+4x-1=0 — step-by-step solution | Explain MAths

-2x^2+4x-1=0

1) Let's solve in

ℝ

:

- 2 x^{2} + 4 x - 1 = 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 (-2) \cdot (-1)

ℹ

4^{2}

=

16

‖

-4 \cdot (-2)

=

8

‖

8 \cdot (-1)

=

-8

Δ = 16 - 8

Δ = 8

Since

Δ > 0

, the equation has two solutions:

For the 1st solution, we obtain:

\frac{- b - \sqrt{Δ}}{2 a}

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

ℹ

- \sqrt{8}

=

- 2 \cdot \sqrt{2}

‖

2 \cdot (-2)

=

-4

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

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

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

Solution 2nd:

\frac{- b + \sqrt{Δ}}{2 a}

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

ℹ

\sqrt{8}

=

2 \cdot \sqrt{2}

‖

2 \cdot (-2)

=

-4

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

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

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

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

Solution found:

x = \frac{2 - \sqrt{2}}{2}

Solution found:

x = \frac{\sqrt{2} + 2}{2}

▶Solution:

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

2) The numeric value of:

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

equals:

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

x = {\frac{\approx 0.5858}{2}, \frac{\approx 3.4142}{2}}

▷

x = {\approx 0.2929, \approx 1.7071}