# How do you solve the equation x^2-10x+25=49 by completing the square?

Dec 13, 2016

$x = 12 \text{ }$ or $\text{ } x = - 2$

#### Explanation:

The difference of squares identity can be written:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Use this with $a = \left(x - 5\right)$ and $b = 7$ later.

Given:

${x}^{2} - 10 x + 25 = 49$

Both sides of this equation are already perfect squares:

${\left(x - 5\right)}^{2} = {x}^{2} - 10 x + 25 = 49 = {7}^{2}$

Subtract ${7}^{2}$ from both ends to get:

$0 = {\left(x - 5\right)}^{2} - {7}^{2} = \left(\left(x - 5\right) - 7\right) \left(\left(x - 5\right) + 7\right) = \left(x - 12\right) \left(x + 2\right)$

Hence:

$x = 12 \text{ }$ or $\text{ } x = - 2$