5.6 Quadratic Equations and Complex Numbers Objectives: Graph and perform operations on complex numbers

Imaginary Numbers A complex number is any number that can be written as a + bi, where a and b are real numbers and a is called the real part and b is called the imaginary part. 3 3 + 4i 4i real part imaginary part

Example 1 Find x and y such that -3x + 4iy = 21 – 16i. Real parts Imaginary parts -3x = 21 4y = -16 y = -4 x = -7 x = -7 and y = -4

Example 2 Find each sum or difference. a) (-10 – 6i) + (8 – i) = (-10 + 8) + (-6i – i) = -2 – 7i b) (-9 + 2i) – (3 – 4i) = (-9 – 3) + (2i + 4i) = -12 + 6i

Example 3 Multiply. (2 – i)(-3 – 4i) = -6 - 8i + 3i + 4i2 = -6 - 5i + 4(-1) = -10 – 5i

Conjugate of a Complex Number The conjugate of a complex number a + bi is a – bi. The conjugate of a + bi is denoted a + bi.

Example 4 (3 – 2i) (-4 – i) = (-4 + i) (-4 - i) -12 - 3i + 8i + 2i2 = multiply by 1, using the conjugate of the denominator (3 – 2i) (-4 – i) = (-4 + i) (-4 - i) -12 - 3i + 8i + 2i2 = 16 + 4i - 4i - i2 -12 + 5i + 2(-1) -14 + 5i = = 16 - (-1) 17

Practice

Warm-Up 5 minutes Perform the indicated operations, and simplify. 1) (-4 + 2i) + (6 – 3i) 2) (2 + 5i) – (5 + 3i) 3) (7 + 7i) – (-6 – 2i) 4)

Warm-Up 6 minutes Use the quadratic formula to solve each equation. 1) x2 + 12x + 35 = 0 2) x2 + 81 = 18x 3) x2 + 4x – 9 = 0 4) 2x2 = 5x + 9

5.6 Quadratic Equations and Complex Numbers Objectives: Classify and find all roots of a quadratic equation

Solutions of a Quadratic Equation The expression b2 – 4ac is called the discriminant. Let ax2 + bx + c = 0, where a = 0. If b2 – 4ac > 0, then the quadratic equation has 2 distinct real solutions. If b2 – 4ac = 0, then the quadratic equation has 1 real solutions. If b2 – 4ac < 0, then the quadratic equation has 0 real solutions.

Example 1 Find the discriminant for each equation. Then determine the number of real solutions. a) 3x2 – 6x + 4 = 0 b2 – 4ac = (-6)2 – 4(3)(4) = 36 – 48 = -12 no real solutions b) 3x2 – 6x + 3 = 0 b2 – 4ac = (-6)2 – 4(3)(3) = 36 – 36 = one real solution c) 3x2 – 6x + 2 = 0 b2 – 4ac = (-6)2 – 4(3)(2) = 36 – 24 = 12 two real solutions

Practice Identify the number of real solutions: 1) -3x2 – 6x + 15 = 0

Imaginary Numbers The imaginary unit is defined as and i2 = -1. If r > 0, then the imaginary number is defined as follows:

Example 2 Solve 6x2 – 3x + 1 = 0.

Practice Solve -4x2 + 5x – 3 = 0.

5 minutes Warm-Up Find the discriminant, and determine the number of real solutions. Then solve. 1) x2 – 7x = -10 2) 5x2 + 4x = -5

5.6.3 Quadratic Equations and Complex Numbers Objectives: Graph and perform operations on complex numbers

The Complex Plane In the complex plane, the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. imaginary axis -4 -2 2 4 real axis