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Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers. The Discriminant The Discriminant Example 1 Example 1 Example 1 Example 1 Try This • Find the discriminant for each equation. Then, determine the number of real solutions. 3x 6 x 15 0 2 2x 4x 3 0 2 Try This • Find the discriminant for each equation. Then, determine the number of real solutions. 3x 6 x 15 0 2 (6) 2 4(3)(15) 216 2 real roots 2x 4x 3 0 2 Try This • Find the discriminant for each equation. Then, determine the number of real solutions. 3x 6 x 15 0 2 2x 4x 3 0 2 (6) 2 4(3)(15) 216 (4) 2 4(2)(3) 8 2 real roots 0 real roots Imaginary Numbers • If the discriminant is negative, that means when using the quadratic formula, you will have a negative number under a square root. This is what we call an imaginary number and is defined as: i 1 i 1 2 Imaginary Numbers 3 1 3 i 3 8 1 4 2 2i 2 45 1 9 5 3i 5 Example 2 Example 2 Try This • Use the quadratic formula to solve: 4 x 2 5x 3 0 Try This • Use the quadratic formula to solve: 4 x 2 5x 3 0 5 (5) 4(4)( 3) 2(4) 2 5 25 48 5 23 5 i 23 8 8 8 8 Example 3 Example 3 Try This • Find x and y such that 2x + 3iy = -8 + 10i Try This • Find x and y such that 2x + 3iy = -8 + 10i 2 x 8 x 4 real part 3iy 10i 3 y 10 y 103 imaginary part Example 4 Example 4 Additive Inverses • Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses. (4 3i ) (4 3i ) 0 Additive Inverses • Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses. (4 3i ) (4 3i ) 0 • What is the additive inverse of 2i – 12? Additive Inverses • Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses. (4 3i ) (4 3i ) 0 • What is the additive inverse of 2i – 12? -2i + 12 Example 5 Example 5 Try This • Multiply (6 4i )(5 4i ) Try This • Multiply (6 4i )(5 4i ) 30 24i 20i 16i 2 30 44i 16(1) 14 44i Conjugate of a Complex Number • In order to simplify a fraction containing complex numbers, you often need to use the conjugate of a complex number. For example, the conjugate of 2 + 5i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i. Conjugate of a Complex Number • In order to simplify a fraction containing complex numbers, you often need to use the conjugate of a complex number. For example, the conjugate of 2 + 5i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i. ________ • The conjugate of a bi is denoted a bi . Conjugate of a Complex Number • In order to simplify a fraction containing complex numbers, you often need to use the conjugate of a complex number. For example, the conjugate of 2 + 5i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i. ________ • The conjugate of a bi is denoted a bi . • To simplify a quotient with an imaginary number, multiply by 1 using the conjugate of the denominator. Example 6 • Simplify form. 2 5i . Write your answer in standard 2 3i Example 6 • Simplify form. 2 5i . Write your answer in standard 2 3i • Multiply the top and bottom by 2 + 3i. 2 5i 2 3i 4 6i 10i 15i 2 11 16i 2 2 3i 2 3i 4 6i 6i 9i 13 13 Example 6 • Simplify form. 3 4i . Write your answer in standard 2i Example 6 • Simplify form. 3 4i . Write your answer in standard 2i • Multiply the top and bottom by 2 – i. 3 4i 2 i 6 3i 8i 4i 2 2 11i 2 2 i 2 i 4 2i 2i i 5 5 Homework • Page 320 • 24-66 multiples of 3