factor 142x^2-228x-112 — step-by-step solution

Let's factor in

ℝ

142 x^{2} - 228 x - 112

We factor out

2

, the greatest common divisor of the coefficients:

142 x^{2} - 228 x - 112

2 \cdot (71 x^{2} - 114 x - 56)

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 71

b = -114

and

c = -56

Δ = {(-114)}^{2} - 4 \cdot 71 \cdot (-56)

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

{(-114)}^{2}

(-114) \cdot (-114)

12,996

■Multiplications are done before additions and subtractions:

4 \cdot 71

4 \cdot (70 + 1)

4 \cdot 70 + 4

280 + 4

284

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

- 284 \cdot (-56)

284 \cdot 56

Δ = 12,996 + 284 \cdot 56

ⓘ Multiplications are done before additions and subtractions:

284 \cdot 56

284 \cdot (50 + 6)

284 \cdot 50 + 284 \cdot 6

14,200 + 1704

15,904

Δ = 12,996 + 15,904

ⓘ Start from

12,996

and move forward

15,904

on the number line:

12,996 + 15,904

28,900

Δ = 28,900

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-114) - \sqrt{28,900}}{2 \cdot 71}

x_{1} = \frac{114 - \sqrt{28,900}}{2 \cdot 71}

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

\sqrt{28,900}

170

;

170^{2}

28,900

ⅰ

2 \cdot 71

2 \cdot (70 + 1)

2 \cdot 70 + 2

140 + 2

142

x_{1} = \frac{114 - 170}{142}

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

114 - 170

-56

x_{1} = - \frac{56}{142}

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

2

, their greatest common divisor:

- \frac{56}{142}

- \frac{28 \cdot 2}{71 \cdot 2}

- \frac{28}{71}

x_{1} = - \frac{28}{71}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-114) + \sqrt{28,900}}{2 \cdot 71}

x_{2} = \frac{114 + \sqrt{28,900}}{2 \cdot 71}

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

\sqrt{28,900}

170

;

170^{2}

28,900

ⅰ

2 \cdot 71

2 \cdot (70 + 1)

2 \cdot 70 + 2

140 + 2

142

x_{2} = \frac{114 + 170}{142}

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

114 + 170

284

x_{2} = \frac{284}{142}

We divide the numerator and denominator of

\frac{284}{142}

by the common factor:

142

, which gives the new expression:

2

x_{2} = 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:

71 \cdot (x - 2) \cdot (x + \frac{28}{71})

(x - 2) \cdot (71 x + 28)

Factors identified:

(x - 2) \cdot (71 x + 28)

▶Factored expression:

2 \cdot (x - 2) \cdot (71 x + 28)