factor 2x^2-9x-18 — step-by-step solution

Let's factor in

ℝ

2 x^{2} - 9 x - 18

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = -9

and

c = -18

Δ = {(-9)}^{2} - 4 \cdot 2 \cdot (-18)

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

{(-9)}^{2}

(-9) \cdot (-9)

81

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

■Two minus signs cancel out, the product of two negative numbers is positive:

- 8 \cdot (-18)

8 \cdot 18

Δ = 81 + 8 \cdot 18

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 18

8 \cdot (10 + 8)

8 \cdot 10 + 8 \cdot 8

80 + 64

144

Δ = 81 + 144

ⓘ Start from

81

and move forward

144

on the number line:

81 + 144

225

Δ = 225

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-9) - \sqrt{225}}{2 \cdot 2}

x_{1} = \frac{9 - \sqrt{225}}{2 \cdot 2}

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

\sqrt{225}

15

;

15^{2}

225

ⅰ

2 \cdot 2

4

x_{1} = \frac{9 - 15}{4}

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

9 - 15

-6

x_{1} = - \frac{6}{4}

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

2

, their greatest common divisor:

- \frac{6}{4}

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

- \frac{3}{2}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-9) + \sqrt{225}}{2 \cdot 2}

x_{2} = \frac{9 + \sqrt{225}}{2 \cdot 2}

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

\sqrt{225}

15

;

15^{2}

225

ⅰ

2 \cdot 2

4

x_{2} = \frac{9 + 15}{4}

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

9 + 15

24

x_{2} = \frac{24}{4}

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

4

, their greatest common divisor:

\frac{24}{4}

6

x_{2} = 6

A trinomial with roots

x_{1}

and

x_{2}

a \cdot (x - x_{1}) \cdot (x - x_{2})

; bringing the coefficient into the brackets:

2 \cdot (x - 6) \cdot (x + \frac{3}{2})

(x - 6) \cdot (2 x + 3)

▶Factors identified:

(x - 6) \cdot (2 x + 3)