factor 32x^2-4x-15 — step-by-step solution

Let's factor in

ℝ

32 x^{2} - 4 x - 15

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 32

b = -4

and

c = -15

Δ = {(-4)}^{2} - 4 \cdot 32 \cdot (-15)

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

{(-4)}^{2}

(-4) \cdot (-4)

16

■Multiplications are done before additions and subtractions:

4 \cdot 32

4 \cdot (30 + 2)

4 \cdot 30 + 4 \cdot 2

120 + 8

128

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

- 128 \cdot (-15)

128 \cdot 15

Δ = 16 + 128 \cdot 15

ⓘ Multiplications are done before additions and subtractions:

128 \cdot 15

128 \cdot (10 + 5)

128 \cdot 10 + 128 \cdot 5

1280 + 640

1920

Δ = 16 + 1920

ⓘ Start from

16

and move forward

1920

on the number line:

16 + 1920

1936

Δ = 1936

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-4) - \sqrt{1936}}{2 \cdot 32}

x_{1} = \frac{4 - \sqrt{1936}}{2 \cdot 32}

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

\sqrt{1936}

44

;

44^{2}

1936

ⅰ

2 \cdot 32

2 \cdot (30 + 2)

2 \cdot 30 + 2 \cdot 2

60 + 4

64

x_{1} = \frac{4 - 44}{64}

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

4 - 44

-40

x_{1} = - \frac{40}{64}

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

8

, their greatest common divisor:

- \frac{40}{64}

- \frac{5 \cdot 8}{8 \cdot 8}

- \frac{5}{8}

x_{1} = - \frac{5}{8}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-4) + \sqrt{1936}}{2 \cdot 32}

x_{2} = \frac{4 + \sqrt{1936}}{2 \cdot 32}

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

\sqrt{1936}

44

;

44^{2}

1936

ⅰ

2 \cdot 32

2 \cdot (30 + 2)

2 \cdot 30 + 2 \cdot 2

60 + 4

64

x_{2} = \frac{4 + 44}{64}

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

4 + 44

48

x_{2} = \frac{48}{64}

We simplify the fraction:

\frac{48}{64}

by the common factor:

16

to get the new quotient:

\frac{3}{4}

x_{2} = \frac{3}{4}

A trinomial with roots

x_{1}

and

x_{2}

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

; bringing the coefficient into the brackets:

32 \cdot (x + \frac{5}{8}) \cdot (x - \frac{3}{4})

(4 x - 3) \cdot (8 x + 5)

▶Factors found:

(4 x - 3) \cdot (8 x + 5)