Simplify each expression.

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Complex Numbers and Roots

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Simplify each expression. Warm Up Simplify each expression. 2. 1. 3. Find the zeros of each function. 4. f(x) = x2 – 18x + 16 5. f(x) = x2 + 8x – 24

Objectives Vocabulary Define and use imaginary and complex numbers. Solve quadratic equations with complex roots. Vocabulary imaginary unit imaginary number complex number real part imaginary part complex conjugate

Example 1A: Simplifying Square Roots of Negative Numbers Express the number in terms of i.

Example 2: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. 5x2 + 90 = 0 9x2 + 25 = 0

A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i = . The set of real numbers is a subset of the set of complex numbers C. Every complex number has a real part a and an imaginary part b.

Example 3: Equating Two Complex Numbers Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true . Real parts 4x + 10i = 2 – (4y)i Imaginary parts

Example 4A: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. f(x) = x2 + 10x + 26 Factoring is NOT a option with complex zeros

Example 4B: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. g(x) = x2 + 4x + 12

The solutions and are related The solutions and are related. These solutions are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a – bi. If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates. When given one complex root, you can always find the other by finding its conjugate. Helpful Hint

Example 5: Finding Complex Zeros of Quadratic Functions Find each complex conjugate. B. 6i A. 8 + 5i HW: pg 97 18-56 e, 58-71, 76-80