3x^2+8x-4=0 — step-by-step solution

1) Let's solve in

ℝ

3 x^{2} + 8 x - 4 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 3

b = 8

and

c = -4

Δ = 8^{2} - 4 \cdot 3 \cdot (-4)

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

8^{2}

8 \cdot 8

64

■Multiplications are done before additions and subtractions:

4 \cdot 3

12

■Two minus signs cancel out, the product of two negative numbers is positive:

- 12 \cdot (-4)

12 \cdot 4

Δ = 64 + 12 \cdot 4

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 4

(10 + 2) \cdot 4

10 \cdot 4 + 2 \cdot 4

40 + 8

48

Δ = 64 + 48

ⓘ Start from

64

and move forward

48

on the number line:

64 + 48

112

Δ = 112

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 8 - \sqrt{112}}{2 \cdot 3}

x_{1} = \frac{- 8 - \sqrt{112}}{2 \cdot 3}

ⓘ We look for the largest square dividing

112

16

4^{2}

, and take it out of the root:

\sqrt{112}

\sqrt{16 \cdot 7}

\sqrt{16} \cdot \sqrt{7}

4 \cdot \sqrt{7}

■The minus sign is factored out, changing the sign of each term:

- 8 - 4 \cdot \sqrt{7}

- (8 + 4 \cdot \sqrt{7})

ⅰ

2 \cdot 3

6

x_{1} = - \frac{8 + 4 \cdot \sqrt{7}}{6}

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

- \frac{8 + 4 \cdot \sqrt{7}}{6}

- \frac{2 \cdot (2 + \sqrt{7})}{3}

x_{1} = - \frac{2 \cdot (2 + \sqrt{7})}{3}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 8 + \sqrt{112}}{2 \cdot 3}

x_{2} = \frac{- 8 + \sqrt{112}}{2 \cdot 3}

ⓘ We look for the largest square dividing

112

16

4^{2}

, and take it out of the root:

\sqrt{112}

\sqrt{16 \cdot 7}

\sqrt{16} \cdot \sqrt{7}

4 \cdot \sqrt{7}

ⅰ

2 \cdot 3

6

x_{2} = \frac{- 8 + 4 \cdot \sqrt{7}}{6}

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

\frac{- 8 + 4 \cdot \sqrt{7}}{6}

\frac{2 \cdot (- 2 + \sqrt{7})}{3}

x_{2} = \frac{2 \cdot (- 2 + \sqrt{7})}{3}

▶Solutions obtained:

x = {- \frac{2 \cdot (2 + \sqrt{7})}{3}, \frac{2 \cdot (\sqrt{7} - 2)}{3}}

2) The numeric value of:

x = {- \frac{2 \cdot (2 + \sqrt{7})}{3}, \frac{2 \cdot (\sqrt{7} - 2)}{3}}

is:

x = {\approx - \frac{2 \cdot (2 + 2.6458)}{3}, \approx \frac{2 \cdot (2.6458 - 2)}{3}}

x = {\approx - \frac{2 \cdot 4.6458}{3}, \approx \frac{2 \cdot 0.6458}{3}}

x = {\approx - \frac{9.2915}{3}, \approx \frac{1.2915}{3}}

▷

x = {\approx -3.0972, \approx 0.4305}