Deriving the Quadratic FormulaTo Derive the quadratic formula we must complete the square.
ax2 + bx + c = 0
First we divide by a:
x2 +(b/a)x + c/a = 0
Then we subtract C/a from both sides:
x2 +(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 = b2/(4a2). Add that to both sides of the equation:
x2 + (b/a)x + (b/(2a))2 = b2/(4a2) - c/a
You might notice that on the left side of the equation I left b2/(4a2) 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 = b2/(4a2) - c/a
To make things a bit prettier, we get a common denominator (damn-nominator (sic)) on the right side:
(x + b/(2a))2 = b2/(4a2) - 4ac/(4a2)
(x + b/(2a))2 = [b2 - 4ac]/(4a2)
Now we take the square root of both sides, and don't forget when you do that you get ±.
Why? Because if you have x2 = 4, the answer is not x = 2, but x = ±2. Now we continue:
x + b/(2a) = ±√[b2 - 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 ±√[b2 - 4ac])/(2a)
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 -2x2 + 16x - 19 = 0 what I do is this:
x = (-b ±√(b2 - 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):
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