factor 100b^2-90b+20 — step-by-step solution

Let's factorize in

ℝ

100 b^{2} - 90 b + 20

We factor out

10

, the greatest common divisor of the coefficients:

100 b^{2} - 90 b + 20

10 \cdot (10 b^{2} - 9 b + 2)

Given the quadratic form

A b^{2} + B b + C

, you can calculate the discriminant:

Δ = B^{2} - 4 A C

. Here

A = 10

B = -9

and

C = 2

Δ = {(-9)}^{2} - 4 \cdot 10 \cdot 2

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

{(-9)}^{2}

(-9) \cdot (-9)

81

■Multiplications are done before additions and subtractions:

4 \cdot 10

40

Δ = 81 - 40 \cdot 2

ⓘ Multiplications are done before additions and subtractions:

40 \cdot 2

80

Δ = 81 - 80

ⓘ Start from

81

and move back

80

on the number line:

81 - 80

1

Δ = 1

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

b_{1}

\frac{- B - \sqrt{Δ}}{2 A}

\frac{- (-9) - \sqrt{1}}{2 \cdot 10}

b_{1} = \frac{9 - 1}{2 \cdot 10}

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

9 - 1

8

ⅰ

2 \cdot 10

20

b_{1} = \frac{8}{20}

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

4

, their greatest common divisor:

\frac{8}{20}

\frac{2 \cdot 4}{5 \cdot 4}

\frac{2}{5}

b_{1} = \frac{2}{5}

The quadratic formula gives

b_{2}

\frac{- B + \sqrt{Δ}}{2 A}

\frac{- (-9) + \sqrt{1}}{2 \cdot 10}

b_{2} = \frac{9 + 1}{2 \cdot 10}

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

9 + 1

10

ⅰ

2 \cdot 10

20

b_{2} = \frac{10}{20}

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

10

, their greatest common divisor:

\frac{10}{20}

\frac{1 \cdot 10}{2 \cdot 10}

\frac{1}{2}

b_{2} = \frac{1}{2}

A trinomial with roots

b_{1}

and

b_{2}

A \cdot (B - B_{1}) \cdot (B - B_{2})

; bringing the coefficient into the brackets:

10 \cdot (b - \frac{1}{2}) \cdot (b - \frac{2}{5})

(2 b - 1) \cdot (5 b - 2)

Factors identified:

(2 b - 1) \cdot (5 b - 2)

▶Factored expression:

10 \cdot (2 b - 1) \cdot (5 b - 2)