SOLUTION: I need help with the following problem, including the steps needed to get to the answer. Thanks so much!! Find the value of c to make {{{g^2-18g+c}}} a perfect square.

Question 273700: I need help with the following problem, including the steps needed to get to the answer. Thanks so much!!
Find the value of c to make a perfect square.
Found 4 solutions by richwmiller, stanbon, solver91311, jim_thompson5910:
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
(18/2)^2=c
(9*2)/2^2=81*4/4=81
c=81

Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
Find the value of c to make a perfect square.
--------------------------------
g^2 - 18g + c
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= g^2 - 18g + (18/2)^2
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g^2 - 18g + 81 is a perfect square
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Ans: c = 81
===============
Cheers,
Stan H.
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!

In general, is a perfect square if and only if

So divide -18 by 2 and square the result.

John

Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Here's the quick way to do it:

Step 1) Take half of the 'g' coefficient -18 to get

Step 2) Square -9 to get

So the value of 'c' is giving us

Here's why it works:

Since you want to be a perfect square, this means you want to be of the form where 'a' is some fixed number. In other words, you want .

Start with the given equation.

FOIL

Take note that the 'g' term on the right side is . So this means that . Solve for 'a' to get

Note: in the step above, notice how we divided by 2. This is why we divided by 2 in the first step in the "quick method" shown above.

Also, notice that the constant term (the term without a 'g' in it) on the right is . So . But since we know that , we can say that .

Note: this explains why we squared the halved value of the middle coefficient.

So this means that giving us the polynomial which is a perfect square since .
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