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**5.6 Quadratic Equations and Complex Numbers**

Objectives: Graph and perform operations on complex numbers

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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

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**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

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**Example 2 Find each sum or difference. a) (-10 – 6i) + (8 – i)**

= ( ) + (-6i – i) = -2 – 7i b) (-9 + 2i) – (3 – 4i) = (-9 – 3) + (2i + 4i) = i

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

+ 4(-1) = -10 – 5i

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**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.

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**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

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Practice

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**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)

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**Warm-Up 6 minutes Use the quadratic formula to solve each equation.**

1) x2 + 12x + 35 = 0 2) x = 18x 3) x2 + 4x – 9 = 0 4) 2x2 = 5x + 9

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**5.6 Quadratic Equations and Complex Numbers**

Objectives: Classify and find all roots of a quadratic equation

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**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.

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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

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

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**Imaginary Numbers The imaginary unit is defined as and i2 = -1.**

If r > 0, then the imaginary number is defined as follows:

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Example 2 Solve 6x2 – 3x + 1 = 0.

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Practice Solve -4x2 + 5x – 3 = 0.

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

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**5.6.3 Quadratic Equations and Complex Numbers**

Objectives: Graph and perform operations on complex numbers

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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

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