Question 88319: Solve by completing the square.
x^2 = 5x + 2
Answer by avilleredbone08(2)   (Show Source):
You can put this solution on YOUR website! First, subtract the x-squared to make it a trinomial.
The goal here is to complete the square on the polynomial
Since the coefficient of the x2 is not a 1, first factor a -1 out of the polynomial.
To divide by -1
divide each term in by -1 term by term.
To divide by 1
The just gets copied along in the numerator.
The answer is
÷ -1 =
To divide x by 1
The x just gets copied along in the numerator.
The answer is x
5x ÷ -1 = -5x
2 ÷ -1 = -2
So, remembering that the -1 has been factored out, we are now working with:
First, let's focus our attention on the -5x term. In particular the coefficient part of it, the -5 . What if we divided this coefficient by 2...
...to get . Now, we square the to get
Second, we'll add the onto our polynomial like this - -2
But wait! We can't just add a to a polynomial like that! That would change the polynomial into an entirely
different polynomial! Remember, one of the goals of completing the square is to
keep the polynomial the same, and just change what it looks like a little bit.
So, third, we'll use a little "trick" here. Since we want to add the onto the polynomial, why don't we just subtract it as well?!? Like this:
-2
Now, all of this looks a little strange--we know. But look what we have done...
...by adding the and then subtracting it again, the net result is that we have added
NOTHING or ZERO (0) to the polynomial, because we know that
- =0. And, adding zero to something doesn't change it; so the polynomial
is the same as it was originally; it just has some extra numbers "floating" around it.
Now, take a look at the part. It can be written like this (x - 5/2)2
And the -2 ? Combine them to get
So, we're done. We've completed the square of and have found that:
= -1 [(x - 5/2)2-33/4]
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