# How do you solve m^2 + 8m + 15 = 0 by completing the square?

Apr 2, 2016

Complete the square to find $m = - 3$ or $m = - 5$

#### Explanation:

Note that:

${\left(m + 4\right)}^{2} = {m}^{2} + 2 \left(m\right) \left(4\right) + {4}^{2} = {m}^{2} + 8 m + 16$

So add $1$ ti both sides of the equation to get:

${m}^{2} + 8 m + 16 = 1$

which we can write as:

${\left(m + 4\right)}^{2} = 1$

Then take the square root of both sides, allowing for both positive and negative square roots to find:

$m + 4 = \pm \sqrt{1} = \pm 1$

Subtract $4$ from both sides to get:

$m = - 4 \pm 1$

That is $m = - 5$ or $m = - 3$

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Alternative method

Use the difference of squares identity:

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

with $a = \left(m + 4\right)$ and $b = 1$ as follows:

$0 = {m}^{2} + 8 m + 15$

$= {\left(m + 4\right)}^{2} - 16 + 15$

$= {\left(m + 4\right)}^{2} - 1$

$= {\left(m + 4\right)}^{2} - {1}^{2}$

$= \left(\left(m + 4\right) - 1\right) \left(\left(m + 4\right) + 1\right)$

$= \left(m + 3\right) \left(m + 5\right)$

So $m = - 3$ or $m = - 5$