factor c^2+6c-27 — step-by-step solution

Let's factorize in

ℝ

c^{2} + 6 c - 27

Given the quadratic form

A c^{2} + B c + C

, you can calculate the discriminant:

Δ = B^{2} - 4 A C

. Here

A = 1

B = 6

and

C = -27

Δ = 6^{2} - 4 \cdot 1 \cdot (-27)

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

6^{2}

6 \cdot 6

36

Δ = 36 - 4 \cdot (-27)

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

- 4 \cdot (-27)

4 \cdot 27

4 \cdot (20 + 7)

4 \cdot 20 + 4 \cdot 7

80 + 28

108

Δ = 36 + 108

ⓘ Start from

36

and move forward

108

on the number line:

36 + 108

144

Δ = 144

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

c_{1}

\frac{- B - \sqrt{Δ}}{2 A}

\frac{- 6 - \sqrt{144}}{2 \cdot 1}

c_{1} = \frac{- 6 - \sqrt{144}}{2}

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

\sqrt{144}

12

;

12^{2}

144

c_{1} = \frac{- 6 - 12}{2}

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

- 6 - 12

-18

c_{1} = - \frac{18}{2}

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

2

, their greatest common divisor:

- \frac{18}{2}

-9

c_{1} = -9

The quadratic formula gives

c_{2}

\frac{- B + \sqrt{Δ}}{2 A}

\frac{- 6 + \sqrt{144}}{2 \cdot 1}

c_{2} = \frac{- 6 + \sqrt{144}}{2}

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

\sqrt{144}

12

;

12^{2}

144

c_{2} = \frac{- 6 + 12}{2}

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

- 6 + 12

6

c_{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

c_{2} = 3

▶Factors found:

(c + 9) \cdot (c - 3)