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

, let's compute 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:

3 \cdot 4

12

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 back

48

on the number line:

64 - 48

16

Δ = 16

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 8 - \sqrt{16}}{2 \cdot 3}

x_{1} = \frac{- 8 - \sqrt{16}}{2 \cdot 3}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{16}

4

;

4^{2}

16

ⅰ

2 \cdot 3

6

x_{1} = \frac{- 8 - 4}{6}

ⓘ The numerator and the denominator are computed before the division:

- 8 - 4

-12

x_{1} = - \frac{12}{6}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

6

, their greatest common divisor:

- \frac{12}{6}

-2

x_{1} = -2

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 8 + \sqrt{16}}{2 \cdot 3}

x_{2} = \frac{- 8 + \sqrt{16}}{2 \cdot 3}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{16}

4

;

4^{2}

16

ⅰ

2 \cdot 3

6

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

ⓘ The numerator and the denominator are computed before the division:

- 8 + 4

-4

x_{2} = - \frac{4}{6}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{4}{6}

- \frac{2 \cdot 2}{3 \cdot 2}

- \frac{2}{3}

x_{2} = - \frac{2}{3}

▶Solutions found:

x = {-2, - \frac{2}{3}}

2) The numeric value of:

x = {-2, - \frac{2}{3}}

equals:

▷

x = {-2, \approx -0.6667}