factor (8x^2+22x+15)/(4x+5) — step-by-step solution

1) Let's factorize in

ℝ

\frac{8 x^{2} + 22 x + 15}{4 x + 5}

We factorize the numerator and the denominator by looking for their roots; those of the denominator will be forbidden values

2) Let's factorize the numerator:

8 x^{2} + 22 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 = 8

b = 22

and

c = 15

Δ = 22^{2} - 4 \cdot 8 \cdot 15

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

22^{2}

22 \cdot 22

484

■Multiplications are done before additions and subtractions:

4 \cdot 8

32

Δ = 484 - 32 \cdot 15

ⓘ Multiplications are done before additions and subtractions:

32 \cdot 15

32 \cdot (10 + 5)

32 \cdot 10 + 32 \cdot 5

320 + 160

480

Δ = 484 - 480

ⓘ Start from

484

and move back

480

on the number line:

484 - 480

4

Δ = 4

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 22 - \sqrt{4}}{2 \cdot 8}

x_{1} = \frac{- 22 - \sqrt{4}}{2 \cdot 8}

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

\sqrt{4}

2

;

2^{2}

4

ⅰ

2 \cdot 8

16

x_{1} = \frac{- 22 - 2}{16}

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

- 22 - 2

-24

x_{1} = - \frac{24}{16}

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

8

, their greatest common divisor:

- \frac{24}{16}

- \frac{3 \cdot 8}{2 \cdot 8}

- \frac{3}{2}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 22 + \sqrt{4}}{2 \cdot 8}

x_{2} = \frac{- 22 + \sqrt{4}}{2 \cdot 8}

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

\sqrt{4}

2

;

2^{2}

4

ⅰ

2 \cdot 8

16

x_{2} = \frac{- 22 + 2}{16}

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

- 22 + 2

-20

x_{2} = - \frac{20}{16}

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

4

, their greatest common divisor:

- \frac{20}{16}

- \frac{5 \cdot 4}{4 \cdot 4}

- \frac{5}{4}

x_{2} = - \frac{5}{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:

8 \cdot (x + \frac{3}{2}) \cdot (x + \frac{5}{4})

(2 x + 3) \cdot (4 x + 5)

▷The numerator

8 x^{2} + 22 x + 15

factors as:

(2 x + 3) \cdot (4 x + 5)

The roots of the denominator are the forbidden values:

x \neq - \frac{5}{4}

3) Let's replace the numerator and the denominator by their factorizations:

\frac{(2 x + 3) \cdot (4 x + 5)}{4 x + 5}

ⓘ The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor

4 x + 5

\frac{(2 x + 3) \cdot (4 x + 5)}{4 x + 5}

2 x + 3

▶Factor identified:

2 x + 3