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 Dependent Variable

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# Imaginary Solutions to Equations

In the complex number system the even-root property can be restated so that x2 = k is equivalent to for any k 0. So an equation such as x2 = -9 that has no real solutions has two imaginary solutions in the complex numbers.

Example 1

Complex solutions to equations

Find the complex solutions to each equation.

a) x2 = -9

b) 3x2 + 2 = 0

Solution

a) First apply the even-root property:

 x2 = -9 x = Â± Even-root property = Â± i = Â± 3i

Check these solutions in the original equation:

 (3i)2 = 9i2 = 9(-1) = -9 (-3i)2 = 9i2 = -9

The solution set is {Â±3i}.

b) First solve the equation for x2: Check these solutions in the original equation. The solution set is .

The basic facts about complex numbers are listed below.

## Complex Numbers

1. Definition of i: , and i2 = -1.

2. A complex number has the form a + bi, where a and b are real numbers.

3. The complex number a + 0i is the real number a.

4. If b is a positive real number, then .

5. The numbers a + bi and a - bi are called complex conjugates of each other. Their product is the real number a2 + b2.

6. Add, subtract, and multiply complex numbers as if they were algebraic expressions with i being the variable, and replace i2 by -1.

7. Divide complex numbers by multiplying the numerator and denominator by the conjugate of the denominator.

8. In the complex number system x2 = k for any real number k is equivalent to x = Â± .