3x^2-3x-60=0 — step-by-step solution

Let's solve in

ℝ

3 x^{2} - 3 x - 60 = 0

Since

3 \neq 0

, divide both sides by

3

x^{2} - x - 20 = 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 = -20

Δ = {(-1)}^{2} - 4 \cdot 1 \cdot (-20)

ⓘ 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 (-20)

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

- 4 \cdot (-20)

80

Δ = 1 + 80

ⓘ Start from

1

and move forward

80

on the number line:

1 + 80

81

Δ = 81

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{81}}{2 \cdot 1}

x_{1} = \frac{1 - \sqrt{81}}{2}

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

\sqrt{81}

9

;

9^{2}

81

x_{1} = \frac{1 - 9}{2}

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

1 - 9

-8

x_{1} = - \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_{1} = -4

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{81}}{2 \cdot 1}

x_{2} = \frac{1 + \sqrt{81}}{2}

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

\sqrt{81}

9

;

9^{2}

81

x_{2} = \frac{1 + 9}{2}

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

1 + 9

10

x_{2} = \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_{2} = 5

▶Solutions found:

x = {-4, 5}