2x^2-6x-3=0 — step-by-step solution

⚠Ⓘ 2x is read as the product 2*x: for a power, write 2^x‖6x is read as the product 6*x: for a power, write 6^x

1) Let's solve in

ℝ

2 x^{2} - 6 x - 3 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

Δ = {(-6)}^{2} - 4 \cdot 2 \cdot (-3)

ℹ

{(-6)}^{2}

36

‖

4 \cdot 2

8

Δ = 36 + 8 \cdot 3

Δ = 36 + 24

Δ = 60

Δ > 0

, the equation has two solutions:

The first solution is:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{6 - \sqrt{60}}{2 \cdot 2}

ℹ

- \sqrt{60}

- 2 \cdot \sqrt{15}

‖

2 \cdot 2

4

x1 = \frac{6 - 2 \cdot \sqrt{15}}{4}

x1 = \frac{3 - \sqrt{15}}{2}

For the second solution, we obtain:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{6 + \sqrt{60}}{2 \cdot 2}

ℹ

\sqrt{60}

2 \cdot \sqrt{15}

‖

2 \cdot 2

4

x2 = \frac{6 + 2 \cdot \sqrt{15}}{4}

x2 = \frac{3 + \sqrt{15}}{2}

Solution found:

x = \frac{3 - \sqrt{15}}{2}

Solution found:

x = \frac{\sqrt{15} + 3}{2}

▶Solution:

x = {\frac{3 - \sqrt{15}}{2}, \frac{\sqrt{15} + 3}{2}}

2) The numeric value of:

x = {\frac{3 - \sqrt{15}}{2}, \frac{\sqrt{15} + 3}{2}}

is:

x = {\frac{3 - \approx 3.8730}{2}, \frac{\approx 3.8730 + 3}{2}}

x = {- \frac{\approx 0.8730}{2}, \frac{\approx 6.8730}{2}}

▷

x = {\approx -0.4365, \approx 3.4365}