### Problems with Multiple Solutions

Problems with multiple solutions on the exam are questions that give you an equation and then ask you how many solutions there are for the equation provided.

These types of questions will usually appear on the algebra section of the test.

Normally, the questions will involve an equation that has a squared number and an integer, although you may also see other types of equations, such as the one in example 3 below.

You will need to consider both positive and negative numbers as potential solutions.

### Multiple Solutions – Practice Questions

Look at these examples.

**Example 1:** How many solutions exist for the following equation?

*x*^{2} + 8 = 0

A. 0

B. 1

C. 2

D. 4

E. 8

**The correct answer is A.**

Remember that any real number squared will always equal a positive number.

Since 8 is added to the first value *x*^{2}, the result will always be 8 or greater.

In other words, since *x*^{2} is always a positive number, the result of the equation would never be 0.

So there are zero solutions for this equation.

**Example 2:** How many solutions exist for the following equation?

*x*^{2} − 9 = 0

F. 0

G. 1

H. 2

I. 4

J. 8

**The correct answer is H.**

As mentioned above, any real number squared will always equal a positive number.

Since 9 is subtracted from *x*^{2}, *x*^{2} needs to be equal to 9.

Both 3 and −3 solve the equation.

So there are two solutions for this equation.

**Example 3:** How many solutions exist for the following equation?

2(*x* + 5) = 14

A. 0

B. 1

C. 2

D. Infinite

E. Cannot be determined

**The correct answer is B.**

Solve the problem as you normally would.

2(*x* + 5) = 14

2*x* + 10 = 14

2*x* = 4

*x* = 2

Then ask yourself if any other solutions are possible.

A negative number or zero would result in the incorrect value on the left side of the equation.

So, there is only one solution to this problem.

In conclusion, the answers for questions like these are that there might be 1, 2, 3, or infinite solutions to the problem provided.

Go back to the **Algebra Questions**.

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