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Derivation of Quadratic Formula
by Ron Kurtus (revised 9 March 2013)
A quadratic equation (ax2 + bx + c = 0) can be solved for x by using the quadratic formula:
x = [−b ± √(b2 − 4ac)]/2a
The derivation of that formula was done by some clever manipulation of the elements in the quadratic equation. The best way to see where the formula came from is by working backwards to establish the quadratic equation. Then you can see the steps to take to derive the quadratic formula.
Questions you may have include:
- What are the steps working backwards?
- What are the steps to get to the formula?
- How do you handle the square root?
This lesson will answer those questions.
Working backwards
Start with the quadratic formula and put it in the form of the quadratic equation:
x = [−b ± √(b2 − 4ac)]/2a
Multiply both sides of the equal sign by 2a
2ax = −b ± √(b2 − 4ac)
Add b to both sides
2ax + b = ± √(b2 − 4ac)
Square both sides of the equal sign
(2ax + b)2 = b2 − 4ac
4a2x2 + 4abx + b2 = b2 − 4ac
Subtract b2 from both sides
4a2x2 + 4abx = − 4ac
Add − 4ac to both sides
4a2x2 + 4abx + 4ac = 0
Divide by 4a
ax2 + bx + c = 0
Start from equation
Knowing the steps, you can start from the quadratic equation to get the formula:
ax2 + bx + c = 0
Multiply both sides of the equal sign by 4a
4a2x2 + 4abx + 4ac = 0
4a2x2 + 4abx = − 4ac
4a2x2 + 4abx + b2 = b2 − 4ac
Factor 4a2x2 + 4abx + b2 to get (2ax + b)2
(2ax + b)2 = b2 − 4ac
Taking the square root
When you get to (2ax + b)2 = b2 − 4ac, you want to take the square root of each side of the equal sign.
Note that both (2ax + b)*(2ax + b) and [−(2ax + b)]*[−(2ax + b)] equal (2ax + b)2.
That means that the square root can be either plus (+) or minus (−). Thus:
x = [−b ± √(b2 − 4ac)]/2a
The derivation is complete.
Summary
You can see the steps to derive the quadratic formula
x = [−b ± √(b2 − 4ac)]/2a by first going backwards to get the quadratic equation (ax2 + bx + c = 0). Then you can reverse the steps to go from the equation to the formula. When you take the square root, you need to realize that plus or minus factors can be used.
Go step by step
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Derivation of Quadratic Formula