3x^2+10x+2=0 — step-by-step solution

1) Let's solve in

ℝ

3 x^{2} + 10 x + 2 = 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 = 10

and

c = 2

Δ = 10^{2} - 4 \cdot 3 \cdot 2

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

10^{2}

10 \cdot 10

100

■Multiplications are done before additions and subtractions:

4 \cdot 3

12

Δ = 100 - 12 \cdot 2

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 2

(10 + 2) \cdot 2

10 \cdot 2 + 2 \cdot 2

20 + 4

24

Δ = 100 - 24

ⓘ Start from

100

and move back

24

on the number line:

100 - 24

76

Δ = 76

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 10 - \sqrt{76}}{2 \cdot 3}

x_{1} = \frac{- 10 - \sqrt{76}}{2 \cdot 3}

ⓘ We look for the largest square dividing

76

4

2^{2}

, and take it out of the root:

\sqrt{76}

\sqrt{4 \cdot 19}

\sqrt{4} \cdot \sqrt{19}

2 \cdot \sqrt{19}

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

- 10 - 2 \cdot \sqrt{19}

- (10 + 2 \cdot \sqrt{19})

ⅰ

2 \cdot 3

6

x_{1} = - \frac{10 + 2 \cdot \sqrt{19}}{6}

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

- \frac{10 + 2 \cdot \sqrt{19}}{6}

- \frac{5 + \sqrt{19}}{3}

x_{1} = - \frac{5 + \sqrt{19}}{3}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 10 + \sqrt{76}}{2 \cdot 3}

x_{2} = \frac{- 10 + \sqrt{76}}{2 \cdot 3}

ⓘ We look for the largest square dividing

76

4

2^{2}

, and take it out of the root:

\sqrt{76}

\sqrt{4 \cdot 19}

\sqrt{4} \cdot \sqrt{19}

2 \cdot \sqrt{19}

ⅰ

2 \cdot 3

6

x_{2} = \frac{- 10 + 2 \cdot \sqrt{19}}{6}

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

\frac{- 10 + 2 \cdot \sqrt{19}}{6}

\frac{- 5 + \sqrt{19}}{3}

x_{2} = \frac{- 5 + \sqrt{19}}{3}

▶Solutions found:

x = {- \frac{5 + \sqrt{19}}{3}, \frac{\sqrt{19} - 5}{3}}

2) The numeric value of:

x = {- \frac{5 + \sqrt{19}}{3}, \frac{\sqrt{19} - 5}{3}}

equals:

x = {\approx - \frac{5 + 4.3589}{3}, \approx \frac{4.3589 - 5}{3}}

x = {\approx - \frac{9.3589}{3}, \approx - \frac{0.6411}{3}}

▷

x = {\approx -3.1196, \approx -0.2137}