We start with [tex]2 x^{2} -20x-53[/tex] and wish to write it as [tex]a(x+d) ^{2} +e[/tex]
First, pull 2 out from the first two terms: [tex]2( x^{2} -10x)-53[/tex]
Let’s look at what is in parenthesis. In the final form this needs to be a perfect square. Right now we have [tex] x^{2} -10x[/tex] and we can obtain -10x by adding -5x and -5x. That is, we can build the following perfect square: [tex] x^{2} -10x+25=(x-5) ^{2} [/tex]
The “problem” with what we just did is that we added to what was given. Let’s put the expression together. We have [tex]2( x-5) ^{2}-53 [/tex] and when we multiply that out it does not give us what we started with. It gives us [tex]2 x^{2} -20x+50-53=2 x^{2} -20x-3[/tex]
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: [tex]2(x-5) ^{2}-53-50 [/tex] which gives us the final expression we seek:
[tex]2(x-5) ^{2}-103 [/tex]
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
