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

Let's factorize in

ℝ

- x^{2} + 3 x - 2

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 = 3

and

c = -2

Δ = 3^{2} - 4 \cdot (-1) \cdot (-2)

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

3^{2}

3 \cdot 3

9

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

-4 \cdot (-1)

4

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

4 \cdot (-2)

- 4 \cdot 2

Δ = 9 - 4 \cdot 2

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 2

8

Δ = 9 - 8

ⓘ Start from

9

and move back

8

on the number line:

9 - 8

1

Δ = 1

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

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

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

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

- 3 - 1

-4

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

2 \cdot (-1)

-2

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

ⓘ The minus signs of the numerator and the denominator cancel out:

\frac{-4}{-2}

\frac{4}{2}

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

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

2

, their greatest common divisor:

\frac{4}{2}

2

x_{1} = 2