# How do you find the value of c that makes x^2+17x+c into a perfect square?

Jan 25, 2017

c = 72.25

#### Explanation:

in complete the square, we should add square of ½ of the coefficient of x to constant term.
Here 17 is the coefficient of x, ½ of 17 is 8.5 and its square is 72.25

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

(x+8.5)^2 = x^2 + 2 (8.5x) + (8.5)^2 = x^2 + 17x + 72.25

Jan 25, 2017

Use the pattern ${\left(x + \sqrt{c}\right)}^{2} = {x}^{2} + 2 \left(\sqrt{c}\right) x + c$
Set the middle term of the pattern equal to middle term of the given equation and then solve for c.

#### Explanation:

Set the middle term of the pattern equal to middle term of the given equation:

$2 \left(\sqrt{c}\right) x = 17 x$

The first step in solving for c is to divide both sides of the equation by 2x:

$\sqrt{c} = \frac{17}{2}$

The next step in solving for c is to square both sides:

$c = \frac{289}{4}$

This is how you find the value of c that makes the quadratic a perfect square.

Some additional information:

If the middle term of the given equation is negative, then use the pattern ${\left(x - \sqrt{c}\right)}^{2} = {x}^{2} - 2 \left(\sqrt{c}\right) x + c$

Let's do an example, given ${x}^{2} - 8 x + c$

Set the middle terms equal:

$- 2 \left(\sqrt{c}\right) x = - 8 x$

Divide both sides by -2x:

$\left(\sqrt{c}\right) = 4$

Square both sides:

$c = 16$

I hope that this helps.