factor 24y^2-80y+56 — step-by-step solution

Let's factorize in

ℝ

24 y^{2} - 80 y + 56

We factor out

8

, the greatest common divisor of the coefficients:

24 y^{2} - 80 y + 56

8 \cdot (3 y^{2} - 10 y + 7)

Given the quadratic form

a y^{2} + b y + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 3

b = -10

and

c = 7

Δ = {(-10)}^{2} - 4 \cdot 3 \cdot 7

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

{(-10)}^{2}

(-10) \cdot (-10)

100

■Multiplications are done before additions and subtractions:

4 \cdot 3

12

Δ = 100 - 12 \cdot 7

ⓘ Multiplications are done before additions and subtractions:

12 \cdot 7

(10 + 2) \cdot 7

10 \cdot 7 + 2 \cdot 7

70 + 14

84

Δ = 100 - 84

ⓘ Start from

100

and move back

84

on the number line:

100 - 84

16

Δ = 16

Δ > 0

, the equation has two roots:

The quadratic formula gives

y_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-10) - \sqrt{16}}{2 \cdot 3}

y_{1} = \frac{10 - \sqrt{16}}{2 \cdot 3}

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

\sqrt{16}

4

;

4^{2}

16

ⅰ

2 \cdot 3

6

y_{1} = \frac{10 - 4}{6}

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

10 - 4

6

y_{1} = \frac{6}{6}

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

6

, their greatest common divisor:

\frac{6}{6}

1

y_{1} = 1

The quadratic formula gives

y_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-10) + \sqrt{16}}{2 \cdot 3}

y_{2} = \frac{10 + \sqrt{16}}{2 \cdot 3}

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

\sqrt{16}

4

;

4^{2}

16

ⅰ

2 \cdot 3

6

y_{2} = \frac{10 + 4}{6}

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

10 + 4

14

y_{2} = \frac{14}{6}

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

2

, their greatest common divisor:

\frac{14}{6}

\frac{7 \cdot 2}{3 \cdot 2}

\frac{7}{3}

y_{2} = \frac{7}{3}

A trinomial with roots

y_{1}

and

y_{2}

a \cdot (y - y_{1}) \cdot (y - y_{2})

; bringing the coefficient into the brackets:

3 \cdot (y - 1) \cdot (y - \frac{7}{3})

(y - 1) \cdot (3 y - 7)

Factors found:

(y - 1) \cdot (3 y - 7)

▶Factored expression:

8 \cdot (y - 1) \cdot (3 y - 7)