factor 15x^2-19x+6 — step-by-step solution

Let's factorize in

ℝ

15 x^{2} - 19 x + 6

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 15

b = -19

and

c = 6

Δ = {(-19)}^{2} - 4 \cdot 15 \cdot 6

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

{(-19)}^{2}

(-19) \cdot (-19)

361

■Multiplications are done before additions and subtractions:

4 \cdot 15

4 \cdot (10 + 5)

4 \cdot 10 + 4 \cdot 5

40 + 20

60

Δ = 361 - 60 \cdot 6

ⓘ Multiplications are done before additions and subtractions:

60 \cdot 6

360

Δ = 361 - 360

ⓘ Start from

361

and move back

360

on the number line:

361 - 360

1

Δ = 1

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-19) - \sqrt{1}}{2 \cdot 15}

x_{1} = \frac{19 - 1}{2 \cdot 15}

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

19 - 1

18

ⅰ

2 \cdot 15

2 \cdot (10 + 5)

2 \cdot 10 + 2 \cdot 5

20 + 10

30

x_{1} = \frac{18}{30}

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

6

, their greatest common divisor:

\frac{18}{30}

\frac{3 \cdot 6}{5 \cdot 6}

\frac{3}{5}

x_{1} = \frac{3}{5}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-19) + \sqrt{1}}{2 \cdot 15}

x_{2} = \frac{19 + 1}{2 \cdot 15}

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

19 + 1

20

ⅰ

2 \cdot 15

2 \cdot (10 + 5)

2 \cdot 10 + 2 \cdot 5

20 + 10

30

x_{2} = \frac{20}{30}

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

10

, their greatest common divisor:

\frac{20}{30}

\frac{2 \cdot 10}{3 \cdot 10}

\frac{2}{3}

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

A trinomial with roots

x_{1}

and

x_{2}

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

; bringing the coefficient into the brackets:

15 \cdot (x - \frac{2}{3}) \cdot (x - \frac{3}{5})

(3 x - 2) \cdot (5 x - 3)

▶Factors found:

(3 x - 2) \cdot (5 x - 3)