Solving Quadratic Equations Using the Square Root
Property
Example
Solve using the Square Root Property: 16x^{2}  24x + 9 = 81
Solution
Step 1 Write the equation in the form x^{2}
= a.
The first and last terms of the
trinomial are perfect squares. 
(4x)^{2}  24x + (3)^{2}
(4x)^{2}  2(4x)(3)x + (3)^{2} 
= 81 = 81 
Since the trinomial has the form
a^{2}  2ab + b^{2}, it is a perfect square
trinomial. 


A perfect square trinomial can
be written as the product of two
identical binomials. 
(4x  3)(4x  3)
(4x  3)^{2} 
= 81 = 81 
Step 2 Use the Square Root
Property. Step 3 Write each answer in simplified form. 


Simplify 81.
To solve for x, add 3 to both sides.
Divide both sides by 4. 
4x  3 = 9 or
4x = 12 or
x = 3 or 
4x  3 = 9 4x = 6

Step 4 Check each answer.
We leave the check to you. 


So, the two solutions of 16x^{2}  24x + 9 = 81 are 3 and
Note:
This equation can also be solved by
factoring. We first write the equation in the
form ax^{2} + bx + c = 0.
16x^{2}  24x + 9 = 81
16x^{2}  24x  72 = 0
8(2x^{2}  3x  9) = 0
8(2x + 3)(x  3) = 0
2x + 3 = 0 or x  3 = 0
