## Deriving the Quadratic FormulaTo Derive the quadratic formula we must complete the square.ax ^{2} + bx + c = 0First we divide by a: x ^{2} +(b/a)x + c/a = 0 Then we subtract C/a from both sides: x ^{2} +(b/a)x = -c/a Now for completing the square. Take 1/2 of the (b/a) and square it: That gives (b/(2a)) ^{2} = b^{2}/(4a^{2}). Add that to both sides of the equation:x ^{2} + (b/a)x + (b/(2a))^{2} = b^{2}/(4a^{2}) - c/a You might notice that on the left side of the equation I left b ^{2}/(4a^{2}) in the form of (b/(2a))^{2}because that makes it easier to factor when you're completing the square, I think. You don't have to do that; it's up to you. Now we factor the left side of the equation: (x + b/(2a)) ^{2} = b^{2}/(4a^{2}) - c/a To make things a bit prettier, we get a common denominator (damn-nominator (sic)) on the right side: (x + b/(2a)) ^{2} = b^{2}/(4a^{2}) - 4ac/(4a^{2}) (x + b/(2a)) ^{2} = [b^{2} - 4ac]/(4a^{2}) Now we take the square root of both sides, and don't forget when you do that you get ±. Why? Because if you have x ^{2} = 4, the answer is not x = 2, but x = ±2. Now we continue:x + b/(2a) = ±√[b ^{2} - 4ac]/(2a) Note that the term 2a is not in the radical. Now all that's left to do is subtract b/(2a) from both sides and we've got the final answer x = (-b ±√[b ^{2} - 4ac])/(2a) Done! ## Hint for Using the Quadratic Formula!To use the quadratic formula, especially when negative numbers are concerned, I always use parentheses to substitute in the numbers.For example to solve the quadratic equation -2x ^{2} + 16x - 19 = 0 what I do is this:x = (-b ±√(b ^{2} - 4ac)/(2a)x = (-( )±√(( ) ^{2} - 4( )( ))/(2( ))x = (-(16)±√((16) ^{2} - 4(-2)(-19))/(2(-2))You'll notice that this helps to elliminate any errors from putting in negative numbers! The final simplified answer is x = 4 ±√(26)/2 Hope this helps you! Remember, math is 90% practice, 10% theory. Practice your homework problems! A YouTube video of a Quadratic Formula Mnemonic Song (to Pop Goes the Weasel): Homepage: www.tutor-homework.com More math tools, links, & tutorials:Math Tools & Links |