Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic...

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.

01563 2 xx 0342 2 xx

Try This

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

2 real roots

01563 2 xx 0342 2 xx

216)15)(3(4)6( 2

Try This

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

2 real roots 0 real roots

01563 2 xx 0342 2 xx

216)15)(3(4)6( 2 8)3)(2(4)4( 2

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:

1i

12 i

Imaginary Numbers

3313 i

222418 i

5359145 i

Example 2

Example 2

Try This

• Use the quadratic formula to solve:

0354 2 xx

Try This

• Use the quadratic formula to solve:

0354 2 xx

)4(2

)3)(4(4)5(5 2

8

23

8

5

8

235

8

48255

i

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

4

82

x

x

310

103

103

y

y

iiy

Example 4

Example 4

Additive Inverses

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

0)34()34( ii

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?

0)34()34( ii

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

0)34()34( ii

Example 5

Example 5

Try This

• Multiply )45)(46( ii

Try This

• Multiply )45)(46( ii

ii

iii

4414)1(164430

16202430 2

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 .bia ________

bia

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.

bia ________

bia

Example 6

• Simplify . Write your answer in standard form. i

i

32

52

Example 6

• Simplify . Write your answer in standard form.

• Multiply the top and bottom by 2 + 3i.

i

i

32

52

13

16

13

11

9664

151064

32

32

32

522

2 i

iii

iii

i

i

i

i

Example 6

• Simplify . Write your answer in standard form. i

i

2

43

Example 6

• Simplify . Write your answer in standard form.

• Multiply the top and bottom by 2 – i.

i

i

2

43

5

11

5

2

224

4836

2

2

2

432

2 i

iii

iii

i

i

i

i

Homework

• Page 320• 24-66 multiples of 3