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Complex Numbers and Roots Simplify each expression. 1. 2. 3. Find the zeros of each function. 4. f(x) = x2 – 18x + 16 5. f(x) = x2 + 8x – 24 Holt McDougal Algebra 2 Complex Numbers and Roots Objectives Define and use imaginary and complex numbers. Solve quadratic equations with complex roots. Holt McDougal Algebra 2 Complex Numbers and Roots Vocabulary imaginary unit imaginary number complex number real part imaginary part complex conjugate Holt McDougal Algebra 2 Complex Numbers and Roots You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as . You can use the imaginary unit to write the square root of any negative number. Holt McDougal Algebra 2 Complex Numbers and Roots Holt McDougal Algebra 2 Complex Numbers and Roots Example 1: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Holt McDougal Algebra 2 Complex Numbers and Roots Example 2: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Holt McDougal Algebra 2 Complex Numbers and Roots Example 3 Express the number in terms of i. Holt McDougal Algebra 2 Complex Numbers and Roots Example 4 Express the number in terms of i. Holt McDougal Algebra 2 Complex Numbers and Roots Example 5 Express the number in terms of i. Holt McDougal Algebra 2 Complex Numbers and Roots Example 6: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. Holt McDougal Algebra 2 Complex Numbers and Roots Example 7: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. 5x2 + 90 = 0 Holt McDougal Algebra 2 Complex Numbers and Roots Example 8 Solve the equation. x2 = –36 Holt McDougal Algebra 2 Complex Numbers and Roots Example 9 Solve the equation. x2 + 48 = 0 Holt McDougal Algebra 2 Complex Numbers and Roots Example 10 Solve the equation. 9x2 + 25 = 0 Holt McDougal Algebra 2 Complex Numbers and Roots 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. Holt McDougal Algebra 2 Complex Numbers and Roots Real numbers are complex numbers where b = 0. Imaginary numbers are complex numbers where a = 0 and b ≠ 0. These are sometimes called pure imaginary numbers. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Holt McDougal Algebra 2 Complex Numbers and Roots Example 11: Finding Complex Zeros of Quadratic Functions Find the zeros of the function by completing the square. f(x) = x2 + 10x + 26 Holt McDougal Algebra 2 Complex Numbers and Roots Example 12: Finding Complex Zeros of Quadratic Functions Find the zeros of the function by completing the square. g(x) = x2 + 4x + 12 Holt McDougal Algebra 2 Complex Numbers and Roots Example 13 Find the zeros of the function. f(x) = x2 + 4x + 13 Holt McDougal Algebra 2 Complex Numbers and Roots Example 14 Find the zeros of the function. g(x) = x2 – 8x + 18 Holt McDougal Algebra 2 Complex Numbers and Roots 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. Helpful Hint When given one complex root, you can always find the other by finding its conjugate. Holt McDougal Algebra 2 Complex Numbers and Roots Example 15: Finding Complex Zeros of Quadratic Functions Find each complex conjugate. A. 8 + 5i Holt McDougal Algebra 2 B. 6i Complex Numbers and Roots Example 16 Find each complex conjugate. A. 9 – i C. –8i Holt McDougal Algebra 2 B.