6x^2+40x=14 — step-by-step solution

1) Let's solve in

ℝ

6 x^{2} + 40 x = 14

Since

2 \neq 0

, divide both sides by

2

3 x^{2} + 20 x = 7

3 x^{2} + 20 x - 7 = 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 = -7

Δ = 20^{2} - 4 \cdot 3 \cdot (-7)

ⓘ 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

■Two minus signs cancel out, the product of two negative numbers is positive:

- 12 \cdot (-7)

12 \cdot 7

Δ = 400 + 12 \cdot 7

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 7

(10 + 2) \cdot 7

10 \cdot 7 + 2 \cdot 7

70 + 14

84

Δ = 400 + 84

ⓘ Start from

400

and move forward

84

on the number line:

400 + 84

484

Δ = 484

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 20 - \sqrt{484}}{2 \cdot 3}

x_{1} = \frac{- 20 - \sqrt{484}}{2 \cdot 3}

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

\sqrt{484}

22

;

22^{2}

484

ⅰ

2 \cdot 3

6

x_{1} = \frac{- 20 - 22}{6}

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

- 20 - 22

-42

x_{1} = - \frac{42}{6}

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

6

, their greatest common divisor:

- \frac{42}{6}

-7

x_{1} = -7

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 20 + \sqrt{484}}{2 \cdot 3}

x_{2} = \frac{- 20 + \sqrt{484}}{2 \cdot 3}

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

\sqrt{484}

22

;

22^{2}

484

ⅰ

2 \cdot 3

6

x_{2} = \frac{- 20 + 22}{6}

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

- 20 + 22

2

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

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

2

, their greatest common divisor:

\frac{2}{6}

\frac{1 \cdot 2}{3 \cdot 2}

\frac{1}{3}

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

▶Solutions obtained:

x = {-7, \frac{1}{3}}

2) The numeric value of:

x = {-7, \frac{1}{3}}

is:

▷

x = {-7, \approx 0.3333}