12x^2+20x+3=0 — step-by-step solution

1) Let's solve in

ℝ

12 x^{2} + 20 x + 3 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 12

b = 20

and

c = 3

Δ = 20^{2} - 4 \cdot 12 \cdot 3

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

20^{2}

20 \cdot 20

400

■Multiplications are done before additions and subtractions:

4 \cdot 12

4 \cdot (10 + 2)

4 \cdot 10 + 4 \cdot 2

40 + 8

48

Δ = 400 - 48 \cdot 3

ⓘ Multiplications are done before additions and subtractions:

48 \cdot 3

(40 + 8) \cdot 3

40 \cdot 3 + 8 \cdot 3

120 + 24

144

Δ = 400 - 144

ⓘ Start from

400

and move back

144

on the number line:

400 - 144

256

Δ = 256

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 20 - \sqrt{256}}{2 \cdot 12}

x_{1} = \frac{- 20 - \sqrt{256}}{2 \cdot 12}

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

\sqrt{256}

16

;

16^{2}

256

ⅰ

2 \cdot 12

2 \cdot (10 + 2)

2 \cdot 10 + 2 \cdot 2

20 + 4

24

x_{1} = \frac{- 20 - 16}{24}

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

- 20 - 16

-36

x_{1} = - \frac{36}{24}

We divide the numerator and denominator of

\frac{36}{24}

by the common factor:

12

, which gives the new fraction:

\frac{3}{2}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 20 + \sqrt{256}}{2 \cdot 12}

x_{2} = \frac{- 20 + \sqrt{256}}{2 \cdot 12}

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

\sqrt{256}

16

;

16^{2}

256

ⅰ

2 \cdot 12

2 \cdot (10 + 2)

2 \cdot 10 + 2 \cdot 2

20 + 4

24

x_{2} = \frac{- 20 + 16}{24}

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

- 20 + 16

-4

x_{2} = - \frac{4}{24}

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

4

, their greatest common divisor:

- \frac{4}{24}

- \frac{1 \cdot 4}{6 \cdot 4}

- \frac{1}{6}

x_{2} = - \frac{1}{6}

▶Solutions found:

x = {- \frac{3}{2}, - \frac{1}{6}}

2) The numeric value of:

x = {- \frac{3}{2}, - \frac{1}{6}}

equals:

▷

x = {-1.5, \approx -0.1667}