x^2-10=-3x — step-by-step solution

Let's solve in

ℝ

x^{2} - 10 = -3 x

x^{2} + 3 x - 10 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 3

and

c = -10

Δ = 3^{2} - 4 \cdot 1 \cdot (-10)

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

3^{2}

3 \cdot 3

9

Δ = 9 - 4 \cdot (-10)

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

- 4 \cdot (-10)

40

Δ = 9 + 40

ⓘ Start from

9

and move forward

40

on the number line:

9 + 40

49

Δ = 49

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 3 - \sqrt{49}}{2 \cdot 1}

x_{1} = \frac{- 3 - \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{- 3 - 7}{2}

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

- 3 - 7

-10

x_{1} = - \frac{10}{2}

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

2

, their greatest common divisor:

- \frac{10}{2}

-5

x_{1} = -5

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 3 + \sqrt{49}}{2 \cdot 1}

x_{2} = \frac{- 3 + \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{- 3 + 7}{2}

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

- 3 + 7

4

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

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

2

, their greatest common divisor:

\frac{4}{2}

2

x_{2} = 2

▶Solutions found:

x = {-5, 2}