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Solving Quadratic Equations by Completing the Square Method
by Ron Kurtus (updated 18 January 2022)
Some quadratic equations are not easily solved by factoring. In such a case, you can also use the completing the square method to solve the equation.
The method involves rearranging the equation and adding a term to both sides of the equal sign in order to make the left side a squared expression. Then by taking the square root, you can get your solutions.
Although completing the square can give you solutions, the method is not as easy or straightforward as using the quadratic formula.
Questions you may have include:
- What is a practical example of this method?
- What is a general derivation of the method?
- How does it relate the quadratic equation formula?
This lesson will answer those questions.
Practical example
In order to illustrate the method, let's start with the quadratic equation 2x2 − 8x − 12 = 0.
Set the constant in the first term equal to 1 by dividing both sides by 2:
x2 − 4x − 6 = 0
Rearrange the equation by adding 6 to both sides of the equal sign:
x2 − 4x = 6
Take 1/2 the second term constant, square it, and add it to both sides. In this case, add (4/2)2 = 22 to both sides of the equation:
x2 − 4x + 22 = 6 + 4
Note that the expression x2 − 4x + 22 can be factored as a square:
(x − 2)(x − 2) = (x − 2)2 = 10
Take the square root of both sides of the equation, remembering that the result can be a positive (+) number or a negative (−) number:
x − 2 = ±√10
Add 2 to both sides to get your answer:
x = 2 ± √10
General derivation
This derivation will go through completing the square method for the general quadratic equation formula:
ax2 + bx + c = 0
Since it is easier to complete the square when first term equals 1, so divide each term by a:
x2 + (b/a)x + c/a = 0
For convenience, let b/a = d and c/a = e:
x2 + dx + e = 0
Subtract e from both sides of the equation:
x2 + dx = −e
Divide d by 2 and square d/2. Then add (d/2)2 to both sides of the equation:
x2 + dx + (d/2)2 = (d/2)2 − e
Note that x2 + dx + (d/2)2 = (x + d/2)2:
(x + d/2)2 = (d/2)2 − e
Take the square root of both sides of the equation:
x + d/2 = ±√[ (d/2)2 − e]
Subtract d/2 from both sides:
x = −d/2 ± √[ (d/2)2 − e]
Extending to quadratic equation formula
The general solution you get by completing the square can be extended into the quadratic equation formula.
Set d = b/a and e = c/a :
x = −b/2a ± √[ (b/2a)2 − c/a]
Multiply the items under the radical (square root) sign by 4a2/4a2:
x = −b/2a ± √[ (b/2a)2(4a2/4a2) − (c/a)(4a2/4a2)]
Simplify:
x = −b/2a ± √ (b2/4a2 − 4ac/4a2)
Combine terms to put everything over 2a:
x = [−b ± √(b2 − 4ac)]/2a
Lo and behold, this is the quadratic equation formula!
Summary
The completing the square method is a way to solve the quadratic equations. You rearrange the equation and add a term to both sides of the equal sign to make the left side a squared expression. Then by taking the square root, you can get your solutions.
Although completing the square can give you solutions, the method is not as easy or straightforward as using the quadratic formula.
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Resources and references
Websites
Completing the Square: Solving Quadratic Equations - Purpleath.com
Completing the Square - MathIsFun.com
Books
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Solving Quadratic Equations by Completing the Square Method