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.

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

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

Imaginary Numbers

Example 2

Example 2

Try This Use the quadratic formula to solve:

Try This Use the quadratic formula to solve:

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 real part 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.

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?

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? -2 + 12i

Example 5

Example 5

Try This Multiply

Try This Multiply

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 is denoted .

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 is denoted . To simplify a quotient with an imaginary number, multiply by 1 using the conjugate of the denominator.

Example 6 Simplify . Write your answer in standard form.

Example 6 Simplify . Write your answer in standard form. Multiply the top and bottom by 2 + 3i.

Example 6 Simplify . Write your answer in standard form.

Example 6 Simplify . Write your answer in standard form. Multiply the top and bottom by 2 – i.

Homework Page 320 24-66 multiples of 3