One of the most confusing methods of solving a quadratic equation is to complete the square. It may seem pointless, but you should learn it for reference in the next two parts. So bear with me here. Take an equation like x² + 8x + 2 = 35. I'll go through it in steps:
1. Set the equation equal to zero: x² + 8x -33 = 0
2. Get rid of your c term: x² + 8x = 33
3. Get rid of the a term if it is more than 1 by dividing: x² + 8x = 33
4. Now you have to make the quadratic a perfect square by adding your needed c term to both sides. You get this by using the expression (b/2)². x² + 8x + (8/2)² = 33 + (8/2)². The quadratic can automatically be factored from x² + 8x +16 to (x + 4)². This is set in the equation: (x + 4)² = 49.
5. At this point you can take the square route of both sides and end up with (x + 4) = ±7.
6. Now solve it like a regular equation, subtracting the b/2 term, 4, and you get x = ±7 - 4, which makes x = {-11, 3}
Here is the simple explanation to completing the square:
1. Set equation to zero: ax² + bx + c = 0
2. Subtract c term: ax² + bx = -c
3. Divide by a and forget about the other side for now: x² + bx + ?
4. Add (b/2)² to get a perfect square.
5. Now you can factor x² + bx + (b/2)² into (x + b/2)²
6. Solve this side to the rest of the equation which you hopefully have kept up with the other stuff done above.
The problems don't always come out nicely, though. Take 4x² + 16x + 9 = 0
Divide by 4 and move the nine over: x² + 4x = -9/4
Add the c term: x² + 4x + 4 = 5/4
Factor and square root both sides: x + 2 = ± Ö(5)/2
Simplify: x = -2 ± Ö(5)/2
One last thing, be sure to look out for square roots of negative numbers. These are unsolvable as of now. Mark them as "No Real Solution". Another way to mark this is with the symbol: Ø. Beware! Evil Mr. Linear will try to trick you with these.
Now you can apply this to ax² + bx + c = 0 and solve for x after all in the derivation of the quadratic formula!