Derive quadratic formula

Start with ax^2 + bx + c=0 Divide the equation by a x^2 + bx/a + c/a = 0 Put c/a on other side x^2 + bx/a = -c/a Add (b/2a)2 to both sides x^2 + bx/a + (b/2a)^2 = -c/a + (b/2a)^2 The left hand side is now in the x2 + 2dx + d2 format, where "d" is "b/2a" So we can re-write it this way: "Complete the Square" (x+b/2a)^2 = -c/a + (b/2a)^2 Now x only appears once and we are making progress. Now Solve For "x" Now we just need to rearrange the equation to leave "x" on the left Start with (x+b/2a)^2 = -c/a + (b/2a)^2 Square root (x+b/2a) = (+-) sqrt(-c/a+(b/2a)^2) Move b/2a to right x = -b/2a (+-) sqrt(-c/a+(b/2a)^2) That is actually solved! But let's simplify it a bit: Multiply right by 2a/2a x = [ -b (+-) sqrt(-(2a)^2 c/a + (2a)^2(b/2a)^2) ] / 2a Simplify: x = [ -b (+-) sqrt(-4ac + b^2) ] / 2a