3x^2+12x-63=0 — step-by-step solution

Let's solve in

ℝ

3 x^{2} + 12 x - 63 = 0

Since

3 \neq 0

, divide both sides by

3

x^{2} + 4 x - 21 = 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 = 4

and

c = -21

Δ = 4^{2} - 4 \cdot 1 \cdot (-21)

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

4^{2}

4 \cdot 4

16

Δ = 16 - 4 \cdot (-21)

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

- 4 \cdot (-21)

4 \cdot 21

4 \cdot (20 + 1)

4 \cdot 20 + 4

80 + 4

84

Δ = 16 + 84

ⓘ Start from

16

and move forward

84

on the number line:

16 + 84

100

Δ = 100

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 4 - \sqrt{100}}{2 \cdot 1}

x_{1} = \frac{- 4 - \sqrt{100}}{2}

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

\sqrt{100}

10

;

10^{2}

100

x_{1} = \frac{- 4 - 10}{2}

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

- 4 - 10

-14

x_{1} = - \frac{14}{2}

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

2

, their greatest common divisor:

- \frac{14}{2}

-7

x_{1} = -7

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 4 + \sqrt{100}}{2 \cdot 1}

x_{2} = \frac{- 4 + \sqrt{100}}{2}

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

\sqrt{100}

10

;

10^{2}

100

x_{2} = \frac{- 4 + 10}{2}

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

- 4 + 10

6

x_{2} = \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_{2} = 3

▶Solutions obtained:

x = {-7, 3}