3x^2+12x+3=0 — step-by-step solution

1) Let's solve in

ℝ

3 x^{2} + 12 x + 3 = 0

Since

3 \neq 0

, divide both sides by

3

x^{2} + 4 x + 1 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 4

and

c = 1

Δ = 4^{2} - 4 \cdot 1 \cdot 1

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

4^{2}

4 \cdot 4

16

Δ = 16 - 4

ⓘ Start from

16

and move back

4

on the number line:

16 - 4

12

Δ = 12

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 4 - \sqrt{12}}{2 \cdot 1}

x_{1} = \frac{- 4 - \sqrt{12}}{2}

ⓘ We look for the largest square dividing

12

4

2^{2}

, and take it out of the root:

\sqrt{12}

\sqrt{4 \cdot 3}

\sqrt{4} \cdot \sqrt{3}

2 \cdot \sqrt{3}

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

- 4 - 2 \cdot \sqrt{3}

- (4 + 2 \cdot \sqrt{3})

x_{1} = - \frac{4 + 2 \cdot \sqrt{3}}{2}

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

- \frac{4 + 2 \cdot \sqrt{3}}{2}

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

- (2 + \sqrt{3})

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

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (2 + \sqrt{3})

- 2 - \sqrt{3}

x_{1} = - 2 - \sqrt{3}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 4 + \sqrt{12}}{2 \cdot 1}

x_{2} = \frac{- 4 + \sqrt{12}}{2}

ⓘ We look for the largest square dividing

12

4

2^{2}

, and take it out of the root:

\sqrt{12}

\sqrt{4 \cdot 3}

\sqrt{4} \cdot \sqrt{3}

2 \cdot \sqrt{3}

x_{2} = \frac{- 4 + 2 \cdot \sqrt{3}}{2}

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

\frac{- 4 + 2 \cdot \sqrt{3}}{2}

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

- 2 + \sqrt{3}

x_{2} = - 2 + \sqrt{3}

▶Solutions found:

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

2) The numeric value of:

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

is:

x = {\approx - 1.7321 - 2, \approx 1.7321 - 2}

▷

x = {\approx -3.7321, \approx -0.2679}