Solving Quadratic Equations by Factoring
Example
Solve by factoring: 25 + 30x = 9x^{2}
Solution
Step 1 Write the equation in the form
ax^{2} + bx + c = 0 Add 9x^{2} to both sides.
Step 2 Factor the polynomial.
The first and last terms of the
trinomial are perfect squares.
The middle term, 30x, is 2(3x)(5).
Since the trinomial has the form
a^{2} + 2ab + b^{2}, it is a perfect

9x^{2} + 30x +
25 (3x)^{2} + 30x + (5)^{2} (3x)^{2}
+ 2(3x)(5) + (5)^{2} 
= 0 = 0 = 0 
A perfect square trinomial can be
written as the product of two
identical binomials. 
(3x + 5)(3x + 5) 
= 0 
Step 3 Use the Zero Product Property. 
3x + 5 = 0 or 3x + 5 
= 0 
Step 4 Solve the resulting equations.
Step 5 Check each answer.
We leave the check to you. 

So, 25 + 30x = 9x^{2} has two equal solutions,
Every quadratic equation has two solutions.
In the previous example, the two solutions are the same,
. We call
a solution of
multiplicity two.
