Question

When the expression $-2x^2-20x-53$ is written in the form $a(x+d)^2+e$, where $a$, $d$, and $e$ are constants, then what is the sum $a+d+e$?

Answer 1

We start with
`2 x^(2) -20x-53` and wish to write it as
`a(x+d) ^(2) +e`

First, pull 2 out from the first two terms:
`2( x^(2) -10x)-53`

Let’s look at what is in parenthesis. In the final form this needs to be a perfect square. Right now we have
`x^(2) -10x` and we can obtain -10x by adding -5x and -5x. That is, we can build the following perfect square:
`x^(2) -10x+25=(x-5) ^(2)`

The “problem” with what we just did is that we added to what was given. Let’s put the expression together. We have
`2( x-5) ^(2)-53` and when we multiply that out it does not give us what we started with. It gives us
`2 x^(2) -20x+50-53=2 x^(2) -20x-3`

So you see our expression is not right. It should have a -53 but instead has a -3. So to correct it we need to subtract another 50.

We do this as follows:
`2(x-5) ^(2)-53-50` which gives us the final expression we seek:


`2(x-5) ^(2)-103`

If you multiply this out you will get the exact expression we were given. This means that:
a = 2
d = -5
e = -103

We are asked for the sum of a, d and e which is 2 + (-5) + (-103) = -106