2x^2-4x=1 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} - 4 x = 1

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

. Here

a = 2

b = -4

and

c = -1

Δ = {(-4)}^{2} - 4 \cdot 2 \cdot (-1)

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

{(-4)}^{2}

(-4) \cdot (-4)

16

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

■Multiplying by -1 gives the opposite:

- 8 \cdot (-1)

8

Δ = 16 + 8

ⓘ Start from

16

and move forward

8

on the number line:

16 + 8

24

Δ = 24

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-4) - \sqrt{24}}{2 \cdot 2}

x_{1} = \frac{4 - \sqrt{24}}{2 \cdot 2}

ⓘ We look for the largest square dividing

24

4

2^{2}

, and take it out of the root:

\sqrt{24}

\sqrt{4 \cdot 6}

\sqrt{4} \cdot \sqrt{6}

2 \cdot \sqrt{6}

ⅰ

2 \cdot 2

4

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

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

\frac{4 - 2 \cdot \sqrt{6}}{4}

\frac{2 - \sqrt{6}}{2}

x_{1} = \frac{2 - \sqrt{6}}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-4) + \sqrt{24}}{2 \cdot 2}

x_{2} = \frac{4 + \sqrt{24}}{2 \cdot 2}

ⓘ We look for the largest square dividing

24

4

2^{2}

, and take it out of the root:

\sqrt{24}

\sqrt{4 \cdot 6}

\sqrt{4} \cdot \sqrt{6}

2 \cdot \sqrt{6}

ⅰ

2 \cdot 2

4

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

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

\frac{4 + 2 \cdot \sqrt{6}}{4}

\frac{2 + \sqrt{6}}{2}

x_{2} = \frac{2 + \sqrt{6}}{2}

▶Solutions obtained:

x = {\frac{2 - \sqrt{6}}{2}, \frac{2 + \sqrt{6}}{2}}

2) The numeric value of:

x = {\frac{2 - \sqrt{6}}{2}, \frac{2 + \sqrt{6}}{2}}

equals:

x = {\approx \frac{2 - 2.4495}{2}, \approx \frac{2 + 2.4495}{2}}

x = {\approx - \frac{0.4495}{2}, \approx \frac{4.4495}{2}}

▷

x = {\approx -0.2247, \approx 2.2247}