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Solving Quadratic Equations by Factoring

Example

Solve by factoring: 25 + 30x = -9x²

Solution

Step 1 Write the equation in the form ax² + bx + c = 0 Add 9x² 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 a² + 2ab + b², it is a perfect	9x² + 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 = -9x² 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.