# How do you find the value of c that makes x^2+5x+c into a perfect square?

Feb 20, 2017

$c = \frac{25}{4}$

#### Explanation:

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Method 1

Note that ${\left(x + k\right)}^{2} = {x}^{2} + 2 k + {k}^{2}$

So if $2 k = 5$ then $k = \frac{5}{2}$ and:

${\left(x + \frac{5}{2}\right)}^{2} = {x}^{2} + 5 x + {\left(\frac{5}{2}\right)}^{2} = {x}^{2} + 5 x + \frac{25}{4}$

So $c = \frac{25}{4}$

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Method 2

${x}^{2} + 5 x + c$

is in the form:

$a {x}^{2} + b x + c$

with $a = 1$ and $b = 5$.

This has discriminant $\Delta$ given by the formula:

$\Delta = {b}^{2} - 4 a c = 25 - 4 c$

So if $\Delta = 0$ (indicating a repeated zero) then $25 - 4 c = 0$ and hence $c = \frac{25}{4}$.