This problem is on one of the SAT practice tests, in the no-calculator section, and has a few things worth discussing.
What is the sum of all values of x that satisfy the equation:
2x2 – 16x + 8 = 0 ?
SOLUTION:
The SAT occasionally includes a problem on “completing the square.” Rather than memorizing the formulas for a problem that is likely to be rare, it may be more useful to understand what is happening and what needs to be done, particularly being adept at Reverse FOIL:
For this problem, we'll get the same answer if we divide out the 2: x2 – 8x + 4 = 0
This is not easily factored, so move the 4 to the other side: x2 – 8x = -4
What needs to be added to the term on the left for it to be a perfect square? x2 – 8x + 16; i.e. (x - 4)2
[You can avoid some trial and error if you remember (b/2a)2 , made simpler when a = 1]
Add the 16 to both sides of the equation: x2 – 8x + 16 = -4 + 16
Reverse FOIL: (x - 4)2 = 12
x – 4 = +/-√12
x = {4 + √12, 4 - √12}
When we add these two solutions, the √12’s cancel out, leaving the answer: 8
ALTERNATIVE #1: Here is the process for completing the square using formulas you would need to remember:
Using the format ax2 + bx + c = 0
For 2x2 – 16x + 8 = 0, initially, a = 2, and b = -16
Reconfigure: 2x2 – 16x = -8
and divide all terms by a: x2 – 8x = -4
Now, a = 1, and b = -8
Add (b/2)2 to both sides: x2 – 8x + 16 = 12
Reverse FOIL: (x – 4)2 = 12
Square root: x – 4 = +/-√12
Solutions: x = 4 +/- 2√3
ALTERNATIVE #2:
And we can use the Quadratic Equation: x = [-b +/- √(b2 – 4ac)]/2a
(It would be very rare to see a problem where you have to know it, but worth knowing just in case.)
Don't forget that you cannot use a calculator on this one.
-(-16)/2(2) +/- [√256 - (4)(2)(8)]/(2)(2) = 4 +/- (√192)/4 = 4 +/- 2√3