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Determining the Number of Solutions

8.3.1 Determining the Number of Solutions

8.3.1 • Determining the Number of Solutions

Warm Up

How many solutions does each equation have? Be prepared to show your reasoning.

1.

x 2 = 9 x 2 = 9

Compare your answer:

Two solutions

Both 3232 and (−3)2(−3)2 equal 9.

2.

x 2 = 0 x 2 = 0

Compare your answer:

One solution

Only 0202 is 0.

3.

x 2 − 1 = 3 x 2 − 1 = 3

Compare your answer:

Two solutions

Add 1 to both sides of the equation. Then, both 2222 and (−2)2(−2)2 equal 4.

4.

2 x 2 = 50 2 x 2 = 50

Compare your answer:

Two solutions

Divide both sides of the equation by 2. Then, both 5252 and (−5)2(−5)2 equal 25.

5.

( x + 1 ) ( x + 1 ) = 0 ( x + 1 ) ( x + 1 ) = 0

Compare your answer:

One solution

Setting each factor equal to zero results in −1 being the only solution.

6.

x ( x − 6 ) = 0 x ( x − 6 ) = 0

Compare your answer:

Two solutions

Setting each factor equal to zero results in 0 and 6 being solutions.

7.

( x − 1 ) ( x − 1 ) = 4 ( x − 1 ) ( x − 1 ) = 4

Compare your answer:

Two solutions

By trial and error, both 3 and −1 make the equation true.

8.

How are the solutions to questions 2 and 5 different from the other solutions?

Compare your answer:

Questions 2 and 5 only have one solution each, while the others have two solutions.

A double root is a root that appears twice in the solution of an algebraic equation.