factor 56x^2+31x+3 — step-by-step solution

Let's factor in

ℝ

56 x^{2} + 31 x + 3

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 56

b = 31

and

c = 3

Δ = 31^{2} - 4 \cdot 56 \cdot 3

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

31^{2}

31 \cdot 31

961

■Multiplications are done before additions and subtractions:

4 \cdot 56

4 \cdot (50 + 6)

4 \cdot 50 + 4 \cdot 6

200 + 24

224

Δ = 961 - 224 \cdot 3

ⓘ Multiplications are done before additions and subtractions:

224 \cdot 3

(200 + 20 + 4) \cdot 3

200 \cdot 3 + 20 \cdot 3 + 4 \cdot 3

600 + 60 + 12

672

Δ = 961 - 672

ⓘ Start from

961

and move back

672

on the number line:

961 - 672

289

Δ = 289

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 31 - \sqrt{289}}{2 \cdot 56}

x_{1} = \frac{- 31 - \sqrt{289}}{2 \cdot 56}

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

\sqrt{289}

17

;

17^{2}

289

ⅰ

2 \cdot 56

2 \cdot (50 + 6)

2 \cdot 50 + 2 \cdot 6

100 + 12

112

x_{1} = \frac{- 31 - 17}{112}

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

- 31 - 17

-48

x_{1} = - \frac{48}{112}

We simplify the fraction:

\frac{48}{112}

by the common factor:

16

to get the new fraction:

\frac{3}{7}

x_{1} = - \frac{3}{7}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 31 + \sqrt{289}}{2 \cdot 56}

x_{2} = \frac{- 31 + \sqrt{289}}{2 \cdot 56}

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

\sqrt{289}

17

;

17^{2}

289

ⅰ

2 \cdot 56

2 \cdot (50 + 6)

2 \cdot 50 + 2 \cdot 6

100 + 12

112

x_{2} = \frac{- 31 + 17}{112}

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

- 31 + 17

-14

x_{2} = - \frac{14}{112}

The fraction:

\frac{14}{112}

simplifies by the common factor:

14

into the new fraction:

\frac{1}{8}

x_{2} = - \frac{1}{8}

A trinomial with roots

x_{1}

and

x_{2}

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

; bringing the coefficient into the brackets:

56 \cdot (x + \frac{1}{8}) \cdot (x + \frac{3}{7})

(7 x + 3) \cdot (8 x + 1)

▶Factors found:

(7 x + 3) \cdot (8 x + 1)