Question

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

Answer 1

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
`(a+b)^2 = a^2+2ab+b^2`

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

Answer 2

Use the pattern `(x + sqrtc)^2 = x^2 + 2(sqrtc)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(sqrtc)x = 17x`

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

`sqrtc = 17/2`

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

`c = 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 `(x - sqrtc)^2 = x^2 - 2(sqrtc)x + c`

Let's do an example, given `x^2 - 8x + c`

Set the middle terms equal:

`-2(sqrtc)x = -8x`

Divide both sides by -2x:

`(sqrtc) = 4`

Square both sides:

`c = 16`

I hope that this helps.