6x^2+40x+50=0 — step-by-step solution

1) Let's solve in

ℝ

6 x^{2} + 40 x + 50 = 0

Since

2 \neq 0

, divide both sides by

2

3 x^{2} + 20 x + 25 = 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 = 20

and

c = 25

Δ = 20^{2} - 4 \cdot 3 \cdot 25

ⓘ 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 3

12

Δ = 400 - 12 \cdot 25

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 25

(10 + 2) \cdot 25

10 \cdot 25 + 2 \cdot 25

250 + 50

300

Δ = 400 - 300

ⓘ Start from

400

and move back

300

on the number line:

400 - 300

100

Δ = 100

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 20 - \sqrt{100}}{2 \cdot 3}

x_{1} = \frac{- 20 - \sqrt{100}}{2 \cdot 3}

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

\sqrt{100}

10

;

10^{2}

100

ⅰ

2 \cdot 3

6

x_{1} = \frac{- 20 - 10}{6}

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

- 20 - 10

-30

x_{1} = - \frac{30}{6}

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

6

, their greatest common divisor:

- \frac{30}{6}

-5

x_{1} = -5

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 20 + \sqrt{100}}{2 \cdot 3}

x_{2} = \frac{- 20 + \sqrt{100}}{2 \cdot 3}

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

\sqrt{100}

10

;

10^{2}

100

ⅰ

2 \cdot 3

6

x_{2} = \frac{- 20 + 10}{6}

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

- 20 + 10

-10

x_{2} = - \frac{10}{6}

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

2

, their greatest common divisor:

- \frac{10}{6}

- \frac{5 \cdot 2}{3 \cdot 2}

- \frac{5}{3}

x_{2} = - \frac{5}{3}

▶Solutions found:

x = {-5, - \frac{5}{3}}

2) The numeric value of:

x = {-5, - \frac{5}{3}}

is:

▷

x = {-5, \approx -1.6667}