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**Quadratic Equations and Complex Numbers**

Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.

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The Discriminant

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The Discriminant

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Example 1

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Example 1

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Example 1

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Example 1

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Try This Find the discriminant for each equation. Then, determine the number of real solutions.

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Try This Find the discriminant for each equation. Then, determine the number of real solutions. 2 real roots

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Try This Find the discriminant for each equation. Then, determine the number of real solutions. 2 real roots real roots

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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:

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Imaginary Numbers

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Example 2

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Example 2

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Try This Use the quadratic formula to solve:

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Try This Use the quadratic formula to solve:

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Example 3

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Example 3

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Try This Find x and y such that 2x + 3iy = i

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**Try This Find x and y such that 2x + 3iy = -8 + 10i**

real part imaginary part

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Example 4

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Example 4

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Additive Inverses Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses.

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Additive Inverses Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses. What is the additive inverse of 2 – 12i?

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Additive Inverses Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses. What is the additive inverse of 2 – 12i? i

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Example 5

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Example 5

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Try This Multiply

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Try This Multiply

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**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 i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i.

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**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 i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i. The conjugate of is denoted

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**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 i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i. The conjugate of is denoted To simplify a quotient with an imaginary number, multiply by 1 using the conjugate of the denominator.

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Example 6 Simplify Write your answer in standard form.

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**Example 6 Simplify . Write your answer in standard form.**

Multiply the top and bottom by 2 + 3i.

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Example 6 Simplify Write your answer in standard form.

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**Example 6 Simplify . Write your answer in standard form.**

Multiply the top and bottom by 2 – i.

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Homework Page 320 24-66 multiples of 3

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Imaginary & Complex Numbers. Once upon a time… -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented.

Imaginary & Complex Numbers. Once upon a time… -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented.

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