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Quadratic Equation Solver

Solve the Quadratic Formula: AX2 + BX + C = 0
A= B= C=
Significant figures for decimal approximation (up to 20)
X =
Or X =

The exact solution (You may need to simplify this):
X =
Additional Info:

Deriving the Quadratic Formula

To 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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