We start with -2x^2+3x-6.
The way I would solve this is by completing the square. The first step for that is to make the coefficient of x^2 1. We do that by factoring out a -2. The equation now looks like this:
-2(x^2-3/2x+3).
From here, we need to find a term which will make the equation factorable. We do that by taking the middle factor, -3/2, and dividing it by 2, making it -3/4. Then we square this, changing it to 9/16.
Now that we found the number which will make thex^2-3/2part of the equation factorable, what do we do with it? I'll tell you what we do with it; we plug it in . But, we can't just put a random number into the equation. I'll show you how we resolve this in a minute.
First we rewrite the equation as -2(x^2-3/2color(red)(+9/16)color(red)(-9/16)+3 ). NOTE we resolved the issue of sticking in a number by subtracting it so that it actually has no effect on the value of the equation.
Anyway, we can now condense -2(x^2-3/2+9/16-9/16+3 ) into -2((x-3/4)^2+39/16).
We are very nearly done, except that we can stil simplify further by multiplying -2 to the 39/16, making it -39/8.
The final answer is -2(x-3/4)^2-39/8=y
