factor -6x^2-13x-6 — step-by-step solution

Let's factor in

ℝ

- 6 x^{2} - 13 x - 6

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = -6

b = -13

and

c = -6

Δ = {(-13)}^{2} - 4 \cdot (-6) \cdot (-6)

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

{(-13)}^{2}

(-13) \cdot (-13)

169

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

-4 \cdot (-6)

24

■The minus sign is placed in front of the product:

24 \cdot (-6)

- 24 \cdot 6

Δ = 169 - 24 \cdot 6

ⓘ Multiplications are done before additions and subtractions:

24 \cdot 6

(20 + 4) \cdot 6

20 \cdot 6 + 4 \cdot 6

120 + 24

144

Δ = 169 - 144

ⓘ Start from

169

and move back

144

on the number line:

169 - 144

25

Δ = 25

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-13) - \sqrt{25}}{2 \cdot (-6)}

x_{1} = \frac{13 - \sqrt{25}}{2 \cdot (-6)}

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

\sqrt{25}

5

;

5^{2}

25

■A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

2 \cdot (-6)

-12

x_{1} = \frac{13 - 5}{-12}

ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{13 - 5}{-12}

- \frac{13 - 5}{12}

x_{1} = - \frac{13 - 5}{12}

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

13 - 5

8

x_{1} = - \frac{8}{12}

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

4

, their greatest common divisor:

- \frac{8}{12}

- \frac{2 \cdot 4}{3 \cdot 4}

- \frac{2}{3}

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