2 5 Complex Numbers and Roots Warm Up

2 -5 Complex Numbers and Roots Warm Up Lesson Presentation Lesson Quiz Holt Mc. Dougal Algebra 2 Algebra 22 Holt Mc. Dougal

2 -5 Complex Numbers and Roots Warm Up Simplify each expression. 1. Find the zeros of each function. 2. f(x) = x 2 – 18 x + 16 Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Objectives Define and use imaginary and complex numbers. Solve quadratic equations with complex roots. Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Vocabulary imaginary unit imaginary number complex number real part imaginary part complex conjugate Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots You can see in the graph of f(x) = x 2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x 2 + 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 Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Example 1 A: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Factor out – 1. Product Property. Simplify. Multiply. Express in terms of i. Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Example 1 B: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Factor out – 1. Product Property. Simplify. Express in terms of i. Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Check It Out! Example 1 a Express the number in terms of i. Factor out – 1. Product Property. Simplify. Express in terms of i. Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Check It Out! Example 1 b Express the number in terms of i. Factor out – 1. Product Property. Simplify. Multiply. Express in terms of i. Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Check It Out! Example 1 c Express the number in terms of i. Factor out – 1. Product Property. Simplify. Multiply. Express in terms of i. Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Example 2 A: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. Take square roots. Express in terms of i. Check x 2 = – 144 (12 i)2 – 144 i 2 – 144(– 1) – 144 Holt Mc. Dougal Algebra 2 x 2 = (– 12 i)2 144 i 2 144(– 1) – 144

2 -5 Complex Numbers and Roots Example 2 B: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. 5 x 2 + 90 = 0 Add – 90 to both sides. Divide both sides by 5. Take square roots. Express in terms of i. Check 5 x 2 + 90 = 0 0 5(18)i 2 +90 0 90(– 1) +90 0 Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Check It Out! Example 2 a Solve the equation. x 2 = – 36 Take square roots. Express in terms of i. Check x 2 = – 36 (6 i)2 36 i 2 36(– 1) Holt Mc. Dougal Algebra 2 – 36 x 2 = – 36 (– 6 i)2 – 36 36 i 2 – 36 36(– 1) – 36

2 -5 Complex Numbers and Roots Check It Out! Example 2 b Solve the equation. x 2 + 48 = 0 x 2 = – 48 Add – 48 to both sides. Take square roots. Express in terms of i. Check x 2 + 48 = + 48 (48)i 2 + 48 48(– 1) + 48 Holt Mc. Dougal Algebra 2 0 0

2 -5 Complex Numbers and Roots Check It Out! Example 2 c Solve the equation. 9 x 2 + 25 = 0 9 x 2 = – 25 Add – 25 to both sides. Divide both sides by 9. Take square roots. Express in terms of i. Holt Mc. Dougal Algebra 2

2 -5 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 Mc. Dougal Algebra 2

2 -5 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 Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Example 3: Equating Two Complex Numbers Find the values of x and y that make the equation 4 x + 10 i = 2 – (4 y)i true. Real parts 4 x + 10 i = 2 – (4 y)i Imaginary parts 4 x = 2 Equate the real parts. Solve for x. Holt Mc. Dougal Algebra 2 10 = – 4 y Equate the imaginary parts. Solve for y.

2 -5 Complex Numbers and Roots Check It Out! Example 3 a Find the values of x and y that make each equation true. 2 x – 6 i = – 8 + (20 y)i Real parts 2 x – 6 i = – 8 + (20 y)i Imaginary parts 2 x = – 8 Equate the real parts. x = – 4 Solve for x. Holt Mc. Dougal Algebra 2 – 6 = 20 y Equate the imaginary parts. Solve for y.

2 -5 Complex Numbers and Roots Check It Out! Example 3 b Find the values of x and y that make each equation true. – 8 + (6 y)i = 5 x – i Real parts – 8 + (6 y)i = 5 x – i Imaginary parts – 8 = 5 x Equate the real parts. Solve for x. Holt Mc. Dougal Algebra 2 Equate the imaginary parts. Solve for y.

2 -5 Complex Numbers and Roots Example 4 A: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. f(x) = x 2 + 10 x + 26 = 0 Set equal to 0. x 2 + 10 x + Rewrite. = – 26 + x 2 + 10 x + 25 = – 26 + 25 (x + 5)2 = – 1 Add to both sides. Factor. Take square roots. Simplify. Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Example 4 B: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. g(x) = x 2 + 4 x + 12 = 0 Set equal to 0. x 2 + 4 x + Rewrite. = – 12 + x 2 + 4 x + 4 = – 12 + 4 (x + 2)2 = – 8 Add to both sides. Factor. Take square roots. Simplify. Holt Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Check It Out! Example 4 a Find the zeros of the function. f(x) = x 2 + 4 x + 13 = 0 Set equal to 0. x 2 + 4 x + Rewrite. = – 13 + x 2 + 4 x + 4 = – 13 + 4 (x + 2)2 = – 9 Add to both sides. Factor. Take square roots. x = – 2 ± 3 i Holt Mc. Dougal Algebra 2 Simplify.

2 -5 Complex Numbers and Roots Check It Out! Example 4 b Find the zeros of the function. g(x) = x 2 – 8 x + 18 = 0 Set equal to 0. x 2 – 8 x + Rewrite. = – 18 + x 2 – 8 x + 16 = – 18 + 16 Add to both sides. Factor. Take square roots. Simplify. Holt Mc. Dougal Algebra 2

2 -5 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 Mc. Dougal Algebra 2

2 -5 Complex Numbers and Roots Example 5: Finding Complex Zeros of Quadratic Functions Find each complex conjugate. B. 6 i A. 8 + 5 i 8 – 5 i Write as a + bi. Find a – bi. Holt Mc. Dougal Algebra 2 0 + 6 i 0 – 6 i Write as a + bi. Find a – bi. Simplify.

2 -5 Complex Numbers and Roots Check It Out! Example 5 Find each complex conjugate. B. A. 9 – i 9 + (–i) Write as a + bi. 9 – (–i) 9+i Find a – bi. C. – 8 i 0 + (– 8)i 0 – (– 8)i 8 i Holt Mc. Dougal Algebra 2 Simplify. Write as a + bi. Find a – bi. Simplify.

2 -5 Complex Numbers and Roots Lesson Quiz 1. Express in terms of i. Solve each equation. 2. 3 x 2 + 96 = 0 3. x 2 + 8 x +20 = 0 4. Find the values of x and y that make the equation 3 x +8 i = 12 – (12 y)i true. 5. Find the complex conjugate of Holt Mc. Dougal Algebra 2