# How do you factor the expression 25y^2- 52y + 27?

May 21, 2018

$\left(25 y - 27\right) \left(y - 1\right)$

#### Explanation:

We need to find the factors that when multiplied it gives $675$ $\left(25 \times 27\right)$ and when added it gives $- 52$.

Multiplying:
$675$ = $3 \times 3 \times 3 \times 5 \times 5$ = $27 \times 25$

$- 52$ = $- 27 - 25$
Hence, $- 27 \times - 25 = 675$

So the factors are: $- 27$ and $- 25$

$25 {y}^{2} - 52 y + 27$

$25 {y}^{2} - 25 y - 27 y + 27$

$25 y \left(y - 1\right) - 27 \left(y - 1\right)$

$\left(25 y - 27\right) \left(y - 1\right)$

$\left(25 y - 27\right) \left(y - 1\right)$
$\left(25 y \times y\right) - 25 y - 27 y + 27$
$25 {y}^{2} - 52 y + 27$

May 21, 2018

$25 {y}^{2} - 52 y + 27 = \left(y - 1\right) \left(25 y - 27\right)$

#### Explanation:

Given:

$25 {y}^{2} - 52 y + 27$

Note that $25 - 52 + 27 = 0$

Hence $y = 1$ is a zero and $\left(y - 1\right)$ a factor.

The leading term of the other factor must be $25 y$ to get $25 {y}^{2}$ in the product and the trailing term must be $- 27$ in order to get $+ 27$ in the product.

So we find:

$25 {y}^{2} - 52 y + 27 = \left(y - 1\right) \left(25 y - 27\right)$