Imaginary Solutions

Quadratic Formula and Imaginary solutions

Take notes

3 ways to find roots or solutions to quadratic equations

*Factor - but this doesn't always work

*Complete the square - this should always work

*Quadratic Equation - always works and fastest for something complex

Solve by substituting

Ex1: 3x² - 8x - 8

ax² +bx +c,

a=3, b=-8, c=-8

(--8 +or- V8² - 4(3)(8))/2(3)

8 + or- V 64 - (12)(-8))/6

(8+or- V64+96 ) / 6

8+V160 / 6 = (8 +or- 12.65)/6

x=20.65/6=3.44 or 8-12.65=-4.65

Multiple Choice

Solve Ex2: 9x² - 8x + 1

a=9, b=-8, c=1

(--8 +or-V(64 - 4(9)(1) ))/18

• .74, .15

• 6.65, 1.35

• .11, .125

• 1, 9

Ex2: 9x² - 8x + 1

a=9, b=-8, c=1

--8 +or-V64 - 4(9)(1) /18

8 +or- V64 - 36 /18

8 +or-V-28 /18

8+or-2V7 /18 Divide all by 2

4+or-V7 /9= 4+2.65 / 9 or 4-2.65 / 9

x=6.65/9 or 1.35/9 =

=.74..........or .15

Ex 3: 9x² - 8x + 4

a=9, b=-8, c=4

(--8 +or- V(64 -4(9)(4)))/18

(8 +or- V(64-72))/18

(8 +or-V-28)/18

8/18 +4V-7/18 or 8/18 - 4V-7/18

4/9 + 2V-7/9 or 4/9 -2V-7/18

 $\sqrt{-7}$  like all negative radicals, is an imaginary number that has very real meaning and application in physics
We can rewrite it as 7i where i= $\sqrt{-1}$ 

Multiple Choice

If  $$\sqrt{-9}$$  = 3i, and $$\sqrt{-16}$$  = 4i, what is  $$\sqrt{-64}$$  ?

• 5i

• -8i

• 8i

• undefined

Multiple Choice

If $$i^2\equiv\sqrt{-1}\ \ \times\ \sqrt{-1\ }\ \ =\ -1$$  

what is 3i x 5i?

• 15i

• $$\sqrt{15}$$  

• -15