2x^2-7x-2=0 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} - 7 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 = {(-7)}^{2} - 4 \cdot 2 \cdot (-2)

ℹ

{(-7)}^{2}

49

‖

4 \cdot 2

8

Δ = 49 + 8 \cdot 2

Δ = 49 + 16

Δ = 65

Since

Δ > 0

, the equation has two solutions:

Solution first:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{7 - \sqrt{65}}{2 \cdot 2}

x1 = \frac{7 - \sqrt{65}}{4}

Solution second:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{7 + \sqrt{65}}{2 \cdot 2}

x2 = \frac{7 + \sqrt{65}}{4}

Solution obtained:

x = \frac{7 - \sqrt{65}}{4}

Solution obtained:

x = \frac{\sqrt{65} + 7}{4}

▶Solution:

x = {\frac{7 - \sqrt{65}}{4}, \frac{\sqrt{65} + 7}{4}}

2) The numeric value of:

x = {\frac{7 - \sqrt{65}}{4}, \frac{\sqrt{65} + 7}{4}}

equals:

x = {\frac{7 + \approx -8.0623}{4}, \frac{\approx 8.0623 + 7}{4}}

x = {- \frac{\approx 1.0623}{4}, \frac{\approx 15.0623}{4}}

▷

x = {\approx -0.2656, \approx 3.7656}