factor 2x^2+11x-21 — step-by-step solution

Let's factorize in

ℝ

2 x^{2} + 11 x - 21

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 = 11

and

c = -21

Δ = 11^{2} - 4 \cdot 2 \cdot (-21)

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

11^{2}

11 \cdot 11

121

■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 (-21)

8 \cdot 21

Δ = 121 + 8 \cdot 21

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 21

8 \cdot (20 + 1)

8 \cdot 20 + 8

160 + 8

168

Δ = 121 + 168

ⓘ Start from

121

and move forward

168

on the number line:

121 + 168

289

Δ = 289

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 11 - \sqrt{289}}{2 \cdot 2}

x_{1} = \frac{- 11 - \sqrt{289}}{2 \cdot 2}

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

\sqrt{289}

17

;

17^{2}

289

ⅰ

2 \cdot 2

4

x_{1} = \frac{- 11 - 17}{4}

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

- 11 - 17

-28

x_{1} = - \frac{28}{4}

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

4

, their greatest common divisor:

- \frac{28}{4}

-7

x_{1} = -7

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 11 + \sqrt{289}}{2 \cdot 2}

x_{2} = \frac{- 11 + \sqrt{289}}{2 \cdot 2}

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

\sqrt{289}

17

;

17^{2}

289

ⅰ

2 \cdot 2

4

x_{2} = \frac{- 11 + 17}{4}

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

- 11 + 17

6

x_{2} = \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_{2} = \frac{3}{2}

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 + 7) \cdot (x - \frac{3}{2})

(x + 7) \cdot (2 x - 3)

▶Factors found:

(x + 7) \cdot (2 x - 3)