factor 5k^2-18k+9 — step-by-step solution

Let's factor in

ℝ

5 k^{2} - 18 k + 9

Given the quadratic form

a k^{2} + b k + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 5

b = -18

and

c = 9

Δ = {(-18)}^{2} - 4 \cdot 5 \cdot 9

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

{(-18)}^{2}

(-18) \cdot (-18)

324

■Multiplications are done before additions and subtractions:

4 \cdot 5

20

Δ = 324 - 20 \cdot 9

ⓘ Multiplications are done before additions and subtractions:

20 \cdot 9

180

Δ = 324 - 180

ⓘ Start from

324

and move back

180

on the number line:

324 - 180

144

Δ = 144

Δ > 0

, the equation has two roots:

The quadratic formula gives

k_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-18) - \sqrt{144}}{2 \cdot 5}

k_{1} = \frac{18 - \sqrt{144}}{2 \cdot 5}

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

\sqrt{144}

12

;

12^{2}

144

ⅰ

2 \cdot 5

10

k_{1} = \frac{18 - 12}{10}

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

18 - 12

6

k_{1} = \frac{6}{10}

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

2

, their greatest common divisor:

\frac{6}{10}

\frac{3 \cdot 2}{5 \cdot 2}

\frac{3}{5}

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

The quadratic formula gives

k_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-18) + \sqrt{144}}{2 \cdot 5}

k_{2} = \frac{18 + \sqrt{144}}{2 \cdot 5}

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

\sqrt{144}

12

;

12^{2}

144

ⅰ

2 \cdot 5

10

k_{2} = \frac{18 + 12}{10}

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

18 + 12

30

k_{2} = \frac{30}{10}

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

10

, their greatest common divisor:

\frac{30}{10}

3

k_{2} = 3

A trinomial with roots

k_{1}

and

k_{2}

a \cdot (k - k_{1}) \cdot (k - k_{2})

; bringing the coefficient into the brackets:

5 \cdot (k - 3) \cdot (k - \frac{3}{5})

(k - 3) \cdot (5 k - 3)

▶Factors found:

(k - 3) \cdot (5 k - 3)