factor (3y-1)^2+7(3y-1)-44 — step-by-step solution

⚠Ⓘ 3y is read as the product 3*y: for a power, write 3^y‖7( is read as the product 7*(...): for a power, write 7^(...)

Let's factor in

ℝ

{(3 y - 1)}^{2} + 7 \cdot (3 y - 1) - 44

ℹ

{(3 y - 1)}^{2}

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

‖

7 \cdot (3 y - 1)

21 y - 7

(3 y - 1) \cdot (3 y - 1) + (21 y - 7) - 44

(3 y - 1) \cdot (3 y - 1) + 21 y - 51

(3 \cdot y \cdot 3 \cdot y

+ 3 y \cdot (-1)

- 1 \cdot 3 y

- 1 \cdot (-1)) + 21 y - 51

ℹ

3 \cdot 3

9

‖

y \cdot y

y^{2}

‖

3 \cdot (-1)

-3

‖

-1 \cdot 3 y

-3 y

‖

-1 \cdot (-1)

1

9 y^{2}

- 3 y

- 3 y

+ 1

+ 21 y

- 51

ℹ

1 - 51

-50

‖

- 3 y - 3 y + 21 y

15 y

9 y^{2} + 15 y - 50

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

Δ = 15^{2} - 4 \cdot 9 \cdot (-50)

ℹ

15^{2}

225

‖

4 \cdot 9

36

Δ = 225 + 36 \cdot 50

Δ = 225 + 1800

Δ = 2025

Δ > 0

, the equation has two solutions:

The first solution is:

\frac{- b - \sqrt{Δ}}{2 a}

y1 = \frac{- 15 - \sqrt{2025}}{2 \cdot 9}

ℹ

- \sqrt{2025}

-45

‖

2 \cdot 9

18

y1 = \frac{- 15 - 45}{18}

y1 = - \frac{60}{18}

y1 = - \frac{10}{3}

For the second solution, we obtain:

\frac{- b + \sqrt{Δ}}{2 a}

y2 = \frac{- 15 + \sqrt{2025}}{2 \cdot 9}

ℹ

\sqrt{2025}

45

‖

2 \cdot 9

18

y2 = \frac{- 15 + 45}{18}

y2 = \frac{30}{18}

y2 = \frac{5}{3}

▶Factors identified:

9 \cdot (y - \frac{5}{3}) \cdot (y + \frac{10}{3})