6x(x-1)=12-7x — step-by-step solution

1) Let's solve in

ℝ

6 x \cdot (x - 1) = 12 - 7 x

6 x \cdot (x - 1) - 12 + 7 x = 0

ⓘ Expand by multiplying each term inside the parentheses:

6 x \cdot (x - 1)

6 x^{2} + 6 \cdot (- x)

6 x^{2}

+ 6 \cdot (- x)

- 12

+ 7 x = 0

ⓘ The minus sign is placed in front of the product:

6 \cdot (- x)

- 6 x

6 x^{2}

- 6 x

- 12

+ 7 x = 0

ⓘ Like terms are grouped by adding their coefficients:

- 6 x + 7 x

x

6 x^{2} + x - 12 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 6

b = 1

and

c = -12

Δ = 1 - 4 \cdot 6 \cdot (-12)

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 6

24

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

- 24 \cdot (-12)

24 \cdot 12

Δ = 1 + 24 \cdot 12

ⓘ Multiplications are done before additions and subtractions:

24 \cdot 12

24 \cdot (10 + 2)

24 \cdot 10 + 24 \cdot 2

240 + 48

288

Δ = 1 + 288

ⓘ Start from

1

and move forward

288

on the number line:

1 + 288

289

Δ = 289

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 1 - \sqrt{289}}{2 \cdot 6}

x_{1} = \frac{- 1 - \sqrt{289}}{2 \cdot 6}

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

\sqrt{289}

17

;

17^{2}

289

ⅰ

2 \cdot 6

12

x_{1} = \frac{- 1 - 17}{12}

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

- 1 - 17

-18

x_{1} = - \frac{18}{12}

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

6

, their greatest common divisor:

- \frac{18}{12}

- \frac{3 \cdot 6}{2 \cdot 6}

- \frac{3}{2}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 1 + \sqrt{289}}{2 \cdot 6}

x_{2} = \frac{- 1 + \sqrt{289}}{2 \cdot 6}

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

\sqrt{289}

17

;

17^{2}

289

ⅰ

2 \cdot 6

12

x_{2} = \frac{- 1 + 17}{12}

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

- 1 + 17

16

x_{2} = \frac{16}{12}

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

4

, their greatest common divisor:

\frac{16}{12}

\frac{4 \cdot 4}{3 \cdot 4}

\frac{4}{3}

x_{2} = \frac{4}{3}

▶Solutions found:

x = {- \frac{3}{2}, \frac{4}{3}}

2) The numeric value of:

x = {- \frac{3}{2}, \frac{4}{3}}

equals:

▷

x = {-1.5, \approx 1.3333}