Solving Quadratic Equations by Completing the Square
Completing the square is another method for solving quadratic equations. We consider a general quadratic equation in which the coefficient of the leading term is not equal to one, in other words:
![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201216_59.png)
When we try to solve a quadratic equation by completing the square, we are looking for an equation of the form:
![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201216_60.png)
where h and d are constants. The completing the square method involves the following steps:
Step 1) Divide all terms by the coefficient of
.
Step 2) Find ![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201217_61.png)
Step 3) Find ![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201217_63.png)
Step 4) Add
to both sides of the equation.
Step 5) Complete the square on the left-hand-side of the equation.
Step 6) Keep the squared term on the left-hand-side and the constants on the right-hand-side.
Step 7) Take the square root of both sides and solve for the variable.
Example: Solve
by completing the square.
Step 1) Divide all terms by 3:
![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201217_67.png)
Step 2) Divide the coefficient of x by 2:
![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201217_68.png)
Step 3) Square the number from Step 2):
![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201217_69.png)
Step 4) Add the number from Step 3) to both sides of the equation:
![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201217_70.png)
Step 5) Complete the square on the left-hand-side:
Step 6) Isolate the squared term on the left-hand-side of the equation:
![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201217_72.png)
Step 7) Take the square root of both sides and solve:
![](https://libapps-ca.s3.amazonaws.com/accounts/146036/images/Snip20201217_73.png)
Check out this presentation by our Math Tutor Dr. Shohreh Rahmati about solving quadratic equations by completing the square.