factor -3x^2+2x+8 — step-by-step solution

Let's factorize in

ℝ

- 3 x^{2} + 2 x + 8

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = -3

b = 2

and

c = 8

Δ = 2^{2} - 4 \cdot (-3) \cdot 8

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

2^{2}

2 \cdot 2

4

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

-4 \cdot (-3)

12

Δ = 4 + 12 \cdot 8

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 8

(10 + 2) \cdot 8

10 \cdot 8 + 2 \cdot 8

80 + 16

96

Δ = 4 + 96

ⓘ Start from

4

and move forward

96

on the number line:

4 + 96

100

Δ = 100

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 2 - \sqrt{100}}{2 \cdot (-3)}

x_{1} = \frac{- 2 - \sqrt{100}}{2 \cdot (-3)}

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

\sqrt{100}

10

;

10^{2}

100

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

2 \cdot (-3)

-6

x_{1} = \frac{- 2 - 10}{-6}

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

\frac{- 2 - 10}{-6}

- (\frac{- 2 - 10}{6})

x_{1} = - (\frac{- 2 - 10}{6})

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

- 2 - 10

-12

x_{1} = - (\frac{-12}{6})

ⓘ Two minus signs cancel out:

- (\frac{-12}{6})

\frac{12}{6}

x_{1} = \frac{12}{6}

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

6

, their greatest common divisor:

\frac{12}{6}

2

x_{1} = 2