How do we solve linear trigonometric equations?

How do we use logarithms?

Pythagorean Identity

Rationalizing Imaginary Denominators

When dealing with fractions with imaginary denominators, one must express this as a complex number in a+bi form.


For example:

17 - 3i
10 + 7i

1) Multiply by the conjugate of the denominator over itself (fraction equals 

17 - 3i   *   (10 - 7i)         
10 + 7i       (10 - 7i)

2)Foil. FOIL the top and FOIL the bottom.

170 - 119i - 30i + 21i²
100 -70i + 70i + 49i²



3)Multiply top by top, bottom by bottom.



170 - 149i + 21i²                 
    100 + 49i²                                

                                                                               
4)Combine the imaginary terms. They (i) 

cancels out in the denominator.


170 - 149i - 21
    100 + 49

5) Combine constant terms on top and bottom.


149 -149i
      149

6)Split apart into a+bi form.

149  -  149   i
149      149



1 - i

Finding the roots of a parabola.

To find the roots of a parabola is fairly easy. Its all about the formulas.

Questions can be asked like this.

Determine the roots of the equation x2−2x−3=0.

*The trick is to see whether an equation is factorable or not.

x^2 - 2x - 3 = 0    Factorable.

(x - 3) (x + 1)      BAM!

Now find the roots.

x - 3 = 0    x + 1 = 0

x = 3        x = -1

The roots of the equation are 3 and -1.

Now if you get an equation that isn`t factorable like this -x^2 + 4x -3 = 0

Use the quadratic formula.

     a = -1, b = 4, c = -3

-----------------------------------------

-4 + or - √ 4^2 - 4(-1)(-3)

_____________________

                                                                     2(-1)

                  

                                                           -4 + or - √ 16 -12

                                                                   _____________________      

                                                                                        -2

                                                            -4 + or - √ 4                       -4 + or - 2                               

                                                            _________                       __________

                                                                     -2                                       -2

-4 + 2                    or                   -4 - 2

 __________________________________________                                                                   

     -2                                                   -2

-2                                                         -6

__                                                      ____

-2                                                         -2

ROOTS are 1 and 3.  DONE!

Systems of equations :$

*Systems of equations is asked when there are two equations. One must use both to find a coordinate; x and y.
*This is called solving linear systems algebraically.


Solve the system of equations 2x - 3y = 23  +3y=23and -x- 6y= -4 
by combining the equations.

1) First eliminate x.

2x - 3y = 23   -x - 6y = -4

Use both 2 and -1, to eliminate x.

1(2x - 3y = 23)   2( -x - 6y = -4)    Distribute.

2x - 3y = 23

-2x - 12y = -8           2)Make sure to eliminate x, one is positive and one is negative.

____________

0x - 15y = 15

     _________

       -15    -15

y = -1                3) Now use y to find x. Plug in -1 to an easy equation.

2x - 3(-1) = 23

2x + 3 = 23

     -3      -3

__________               

2x = 20

______

    2

x = 10

(10,-1)

Done!

Simplifying Expressions with Rational Exponents =}

Rational exponents can be turned into radicals.

x^1/2 = √x
x^1/3 = ³√x
x^1/4 = ⁴√x

Examples:

1) 27^1/3 = ³√27 = ³√3^3 = 3

2) 12^1/2 * 3^1/2 = √12 * √3 = √36 = 6

3) 16^1/3 * 4^1/3 = ³√16  * ³√4 = ³√64 = 4                

Remember Simplifying:

√2² x² y² z²  =  2xyz

x³ x ^{4 =} x<sup>7      

More Examples: 

1) (x^4 y ^6 z^8) ^1/2
x^4/2 y^6/2 z^8/2</sup>
x² y³ z <sup>4

2) 4^3/2
(4^1/2)^3
(2)^3</sup>
 8

3) (4^3)^1/2

64^1/2
√64
8