Complex numbers
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This document provides a lesson on complex numbers and using imaginary numbers to solve quadratic equations with complex roots. It begins with examples of simplifying expressions involving square roots of negative numbers by expressing them in terms of the imaginary unit i. Then it shows how to solve quadratic equations with imaginary solutions by taking square roots and expressing the solutions in terms of i. Examples are also provided of finding the complex zeros of quadratic functions by setting the functions equal to 0 and factoring. The document emphasizes that complex solutions come in conjugate pairs, and gives examples of finding the complex conjugate of solutions.
Complex numbers
- 1. Complex Numbers and Roots Warm Up Lesson Presentation Lesson Quiz
- 2. Warm Up Simplify each expression. 1. 2. 3. Find the zeros of each function. 4. f(x) = x2 – 18x + 16 5. f(x) = x2 + 8x – 24
- 3. Objectives Define and use imaginary and complex numbers. Solve quadratic equations with complex roots.
- 4. Vocabulary imaginary unit imaginary number complex number real part imaginary part complex conjugate
- 5. 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.
- 6. Example 1A: Simplifying Square Example 1B: Simplifying Square Roots of Roots of Negative Numbers Negative Numbers Express the number in terms of i. Express the number in terms of i. 4 6i 4i 6
- 7. Check It Out! Example 1a Check It Out! Example 1c Express the number in terms of i. Express the number in terms of i. Factor out –1. Product Property. Factor out –1. Product Property. Product Property. Simplify. Simplify. Express in terms of i. Multiply. Example 2A: Solving a Quadratic Equation with Imaginary Solutions Express in terms of i. Solve the equation. Take square roots. Express in terms of i. Check x2 = –144 x2 = –144 (12i)2 –144 (–12i)2 –144 144i 2 –144 144i 2 –144 144(–1) –144 144(–1) –144
- 8. Example 2B: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. 5x2 + 90 = 0 Add –90 to both sides. Divide both sides by 5. Take square roots. Express in terms of i. Check 5x2 + 90 = 0 0 5(18)i 2 +90 0 90(–1) +90 0
- 9. Check It Out! Example 2a Solve the equation. x2 = –36 Take square roots. Express in terms of i. Check x2 = –36 x2 = –36 (6i)2 –36 (–6i)2 –36 36i 2 –36 36i 2 –36 36(–1) –36 36(–1) –36
- 10. Check It Out! Example 2b Solve the equation. x2 + 48 = 0 x2 = –48 Add –48 to both sides. Take square roots. Express in terms of i. Check x2 + 48 = 0 + 48 0 (48)i 2 + 48 0 48(–1) + 48 0
- 11. Check It Out! Example 2c Solve the equation. 9x2 + 25 = 0 9x2 = –25 Add –25 to both sides. Divide both sides by 9. Take square roots. Express in terms of i.
- 12. 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. 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. 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 = 2 Equate the real parts. 10 = –4y Equate the imaginary parts. 4x + 10i = 2 – (4y)i Solve for x. Solve for y. Imaginary parts
- 13. Check It Out! Example 3a Find the values of x and y that make each equation true. 2x – 6i = –8 + (20y)i Real parts 2x – 6i = –8 + (20y)i Imaginary parts Equate the Equate the 2x = –8 –6 = 20y real parts. imaginary parts. x = –4 Solve for x. Solve for y.
- 14. Check It Out! Example 3b Find the values of x and y that make each equation true. –8 + (6y)i = 5x – i 6 Real parts –8 + (6y)i = 5x – i 6 Imaginary parts Equate the –8 = 5x Equate the real imaginary parts. parts. Solve for y. Solve for x.
- 15. Example 4A: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. f(x) = x2 + 10x + 26 x2 + 10x + 26 = 0 Set equal to 0. x2 + 10x + = –26 + Rewrite. x2 + 10x + 25 = –26 + 25 Add to both sides. (x + 5)2 = –1 Factor. Take square roots. Simplify.
- 16. Example 4B: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. g(x) = x2 + 4x + 12 x2 + 4x + 12 = 0 Set equal to 0. x2 + 4x + = –12 + Rewrite. x2 + 4x + 4 = –12 + 4 Add to both sides. (x + 2)2 = –8 Factor. Take square roots. Simplify.
- 17. Check It Out! Example 4a Find the zeros of the function. f(x) = x2 + 4x + 13 x2 + 4x + 13 = 0 Set equal to 0. x2 + 4x + = –13 + Rewrite. x2 + 4x + 4 = –13 + 4 Add to both sides. (x + 2)2 = –9 Factor. Take square roots. x = –2 ± 3i Simplify.
- 18. Check It Out! Example 4b Find the zeros of the function. g(x) = x2 – 8x + 18 x2 – 8x + 18 = 0 Set equal to 0. x2 – 8x + = –18 + Rewrite. x2 – 8x + 16 = –18 + 16 Add to both sides. Factor. Take square roots. Simplify.
- 19. 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. Example 5: Finding Complex Zeros of Quadratic Functions Find each complex conjugate. A. 8 + 5i C. 9 - i 8 + 5i Write as a + bi. 9 + (-i) Write as a + bi. 8 – 5i Find a – bi. 9 – (-i) Find a – bi. 9+i Simplify. B. 6i D. -8i 0 + 6i Write as a + bi. 0 + (-8)i Write as a + bi. 0 – 6i Find a – bi. 0 – (-8)i Find a – bi. –6i Simplify. 8i Simplify.
- 20. Lesson Quiz 1. Express in terms of i. Solve each equation. 2. 3x2 + 96 = 0 3. x2 + 8x +20 = 0 4. Find the values of x and y that make the equation 3x +8i = 12 – (12y)i true. 5. Find the complex conjugate of