x^2-x-10=2 — step-by-step solution

Let's solve in

ℝ

x^{2} - x - 10 = 2

ⓘ We subtract

2

from both sides

x^{2}

- x

- 10

- 2 = 0

ⓘ Start from

-10

and move back

2

more on the number line, away from zero:

- 10 - 2

-12

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 = 1

b = -1

and

c = -12

Δ = {(-1)}^{2} - 4 \cdot 1 \cdot (-12)

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

{(-1)}^{2}

(-1) \cdot (-1)

1

Δ = 1 - 4 \cdot (-12)

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-12)

4 \cdot 12

4 \cdot (10 + 2)

4 \cdot 10 + 4 \cdot 2

40 + 8

48

Δ = 1 + 48

ⓘ Start from

1

and move forward

48

on the number line:

1 + 48

49

Δ = 49

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{49}}{2 \cdot 1}

x_{1} = \frac{1 - \sqrt{49}}{2}

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

\sqrt{49}

7

;

7^{2}

49

x_{1} = \frac{1 - 7}{2}

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

1 - 7

-6

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

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

2

, their greatest common divisor:

- \frac{6}{2}

-3

x_{1} = -3

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{49}}{2 \cdot 1}

x_{2} = \frac{1 + \sqrt{49}}{2}

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

\sqrt{49}

7

;

7^{2}

49

x_{2} = \frac{1 + 7}{2}

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

1 + 7

8

x_{2} = \frac{8}{2}

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

2

, their greatest common divisor:

\frac{8}{2}

4

x_{2} = 4

▶Solutions obtained:

x = {-3, 4}