Solving Quadratic Equations by Completing the
Square
Example Solve by completing the square: 3x^{2}  30 = 21x
Solution
Step 1 Isolate the x^{2}term and the xterm
on one side of the equation.
Add 21x to both sides.
Add 30 to both sides. 
3x^{2}  30 =
21x
3x^{2} + 21x  30 = 0
3x^{2} + 21x = 30 
Step 2 If the coefficient of x^{2} is not 1,
divide both sides of the equation
by the coefficient of x^{2}.
Divide both sides by 3. 
x^{2} + 7x = 10 
Step 3 Find the number that completes
the square: Multiply the coefficient
of x by
. Square the result.
The coefficient of the xterm is 7.


Step 4 Add the result of Step 3 to both
sides of the equation.
Add
to both sides.
Simplify the right side.
The result is
. 

Step 5 Write the trinomial as the
square of a binomial.


Step 6 Finish solving using the
Square Root Property. Use the Square Root Property. 

For each equation, subtract
from both sides and simplify
the radical. 

Step 7 Check each solution.
We leave the check for you. 

The solutions of 3x^{2}  30 = 21x are
Note:
To divide both sides of an equation by 3,
we divide each term by 3, like this:
We can use the â€œplus or minusâ€ symbol to
write the solutions like this
