Question

How do you find the c that makes the trinomial `x^2+9x+c` a perfect square?

Answer 1

`c=81/4`

Explanation:

Note that:

`(x+b/2)^2 = x^2+bx+b^2/4`

So if `b=9` then:

`c = b^2/4 = color(blue)(9)^2/4 = 81/4`

`color(white)()`
Bonus

More generally, we find:

`ax^2+bx+c = a(x+b/(2a))^2+(c-b^2/(4a))`

which if `a=1` simplifies to:

`x^2+bx+c = (x+b/2)^2+(c-b^2/4)`

as in our example.

Answer 2

The pattern is:

`(x+a)^2=x^2+2ax+a^2`

Match the terms of the given equation with the right side of the pattern.

Explanation:

Match the terms of the given equation, `x^2+9x+c` with the right side of the pattern.

Matching the first terms:

`x^2=x^2` does not help us.

Matching the second terms gives us the following equation:

`2ax=9x`

We can use the above equation to solve for the value of "a" by dividing both sides of the equation by 2x:

`a=9/2`

Matching the constant terms, gives us the following equation:

`a^2 = c`

Substitute `9/2` for "a":

`(9/2)^2 = c`

This is your answer:

`c = 81/4`