3x^2=2x+3 — step-by-step solution

1) Let's solve in

ℝ

3 x^{2} = 2 x + 3

3 x^{2} - 2 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 = {(-2)}^{2} - 4 \cdot 3 \cdot (-3)

ℹ

{(-2)}^{2}

4

‖

4 \cdot 3

12

Δ = 4 + 12 \cdot 3

Δ = 4 + 36

Δ = 40

Δ > 0

, the equation has two solutions:

For the 1st solution, we obtain:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{2 - \sqrt{40}}{2 \cdot 3}

ℹ

- \sqrt{40}

- 2 \cdot \sqrt{10}

‖

2 \cdot 3

6

x1 = \frac{2 - 2 \cdot \sqrt{10}}{6}

x1 = \frac{1 - \sqrt{10}}{3}

Solution 2nd:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{2 + \sqrt{40}}{2 \cdot 3}

ℹ

\sqrt{40}

2 \cdot \sqrt{10}

‖

2 \cdot 3

6

x2 = \frac{2 + 2 \cdot \sqrt{10}}{6}

x2 = \frac{1 + \sqrt{10}}{3}

Solution obtained:

x = \frac{1 - \sqrt{10}}{3}

Solution found:

x = \frac{\sqrt{10} + 1}{3}

▶Solution:

x = {\frac{1 - \sqrt{10}}{3}, \frac{\sqrt{10} + 1}{3}}

2) The numeric value of:

x = {\frac{1 - \sqrt{10}}{3}, \frac{\sqrt{10} + 1}{3}}

equals:

x = {\frac{1 + \approx -3.1623}{3}, \frac{\approx 3.1623 + 1}{3}}

x = {- \frac{\approx 2.1623}{3}, \frac{\approx 4.1623}{3}}

▷

x = {\approx -0.7208, \approx 1.3874}