# How do you factor x^2-5x+6?

Mar 13, 2018

take the product to be 6 and sum to be -5
so you will get the factors as $- 2$ and $- 3$
so you take like
${x}^{2} - 2 x - 3 x + 6$
then take the common factor from any two terms
$x \left(x - 2\right) - 3 \left(x - 3\right)$
$\Rightarrow \left(x - 2\right) \left(x - 3\right)$
$x = 2$ & $x = 3$ are the factors of this equation

Mar 13, 2018

Use middle term splitting.
${x}^{2} - 5 x + 6 = \left(x - 2\right) \left(x - 3\right)$

#### Explanation:

Consider the given equation: ${x}^{2} - 5 x + 6$

Here
sum = -5
product = 6

Consider the pair of numbers which when multiplied gives a product (6) and sum (-5).

Since the product is positive, either both numbers should be positive or both numbers should be negative.

The options are:
$1 \cdot 6 = 6$ but $1 + 6 = 7$ (Not required pair)
$\left(- 1\right) \cdot \left(- 6\right) = 6$ but $\left(- 1\right) + \left(- 6\right) = - 7$(Not required pair)
$2 \cdot 3 = 6$ but $2 + 3 = 5$ (Not required pair)
$\left(- 2\right) \cdot \left(- 3\right) = 6$ also $\left(- 2\right) \cdot \left(- 3\right) = - 5$ (Required pair)

So,
${x}^{2} - 5 x + 6 = \left({x}^{2} - 2 x - 3 x + 6\right)$

Now, group the terms.
Then,
${x}^{2} - 5 x + 6 = \left({x}^{2} - 2 x\right) - \left(3 x - 6\right)$
$= x \left(x - 2\right) - 3 \left(x - 2\right)$
$= \left(x - 2\right) \left(x - 3\right)$