**Prerequisite 4a - Solving Equations Algebraically and Graphically**

**I. Vocabulary:**

**A. An Equation**- is a statement that two algebraic expressions are equal.

**Example**: 2x + 4 = 10

**B. To Solve**an equation in "x" means to find all values of "x" for which the statement is true. These values are solutions.

**Example:**2x + 4 = 10

2x = 6

x = 3

**C. An identity equation**- an equation that is true for every real number in the domain of the variable.

**Example:**x

^{2}- 6x + 9 = (x - 3)

^{2}

**D. Conditional Equation**- an equation that is true for just some (or even none of the real numbers in the domain of the variable.

**Example 1:**x

^{2}- 6x + 9 = 0

(x - 3)

^{2}= 0

x = 3 only

**Example 2:**3x + 2 = 4x - 5

x = 7 only

**Example 3:**x

^{2}+ 3x + 4 = 4x - 5

no solutions

**E. Notations:**

1. all real numbers - R

2. no solutions - { } empty set or Æ is the null set

**F. Solving an Equation Involving Fractions:**

Example 1:x/4 + (2x)/3 = 6

Example 1:

multiply by the LCD

4, 3 are the denominators so the LCD = 12, so multiply each term by 12:

12(x/4) + (12)(2x)/3 = (12)(6)

3x + 8x = 72

11x = 72

x = 72/11

**Example 2:**5/x + (3x)/2 = 7

LCD = 2x

(2x)(5/x) + (2x)(3x)/2 = (2x)(7)

10 + 3x

^{2}= 14x

3x

^{2}- 14 x + 10 = 0

use the quadratic equation

x = (-b ± Ö(b

^{2}- 4ac))/(2a)

x = (14 ± Ö(14

^{2}- (4)(3)(10)))/((2)(3))

x = (14 ± Ö76)/3

x = (7 + Ö 19)/3 and x = (7 - Ö19)/3

or

x = 3.786299648 and x = .8803670188 (check to make sure they both work - they do!)

You can also check by using your graphing calculator -

put in

y

_{1}= 5/x + (3x)/2 and

y

_{2}= 7

calc ® intersect, shows the same answers!

**G. Extraneous Solutions - an answer or solution that does not satisfy the original equation.**

**Example:**

6/x - 2/(x+3) = (3(x+5))/(x(x + 3))

LCD = x(x+3)

6(x + 3) - 2x = 3(x + 5)

6x + 18 - 2x = 3x + 15

4x + 18 = 3x + 15

x = -3

substituting x=-3 back into the original equation, we have

-2 - 2/0 = 6/0

This is impossible so there is not a solution

**H. To find the x-intercepts (a, 0), let y = 0 and solve for x.**

**To find the y-intercepts (0, b), let x = 0 and solve for y.**

**Example:**

2x

^{2}- 5x + 2 = y

**x-intercepts, let y = 0**

2x

^{2}- 5x + 2 = 0

(2x -1)(x - 2) = 0

2x - 1 = 0 and x - 2 = 0

x = 1/2 and x = 2

(.5, 0) and (2, 0)

**y-intercepts, let x = 0**

2(0)

^{2}- 5(0) + 2 = y

2 = y

(0,2)

**I. Finding Solutions Graphically:**

24x

^{3}- 36x + 17 = 0

graph this on your calculator and you see there is only one solution:

window:

x-values: -10 to 10

y-values: -5 to 40

You can see it is hard to tell how many times the equation crosses the x-axis

so change the y-values to -3 to 3 and you see it crosses y=0 only once

x = -1.414486, y=0

**J. Can find answers more than one way, here are a couple using the calculator:**

1. y

_{1}= 24x

^{3}- 36x + 17

2nd trace ® zero

enter left-bound before x=-2 and then right-bound after x=-1, then guess about -1.5, gives the answer.

2. y

_{1}= 24x

^{3}- 36x + 17

y

_{2}= 0

2nd trace, #5 intersect, enter 3 times about where the two equations intersect.

**K. Remember with points of intersection, always look for all solutions:**

Example:24x

Example:

^{3}- 36x + 17 = 2x + 5 has how many solutions?

(-1.393598, 2.212803)

(.3407855, 5.681571)

(1.052813, 7.105626)

remember there can be no points of intersection (therefore no solutions), one solution or many solutions.

**L. Solving Polynomial Equations Algebraically:**

1. First degree equation - linear equation -

example: 2x + 4 = 7

2. Second degree equation - quadratic equation -

example: 2x

^{2}+ 4x + 6 = 0

3. Third degree equation - Cubic equation -

example: 2x

^{3}+ 2x

^{2}+ 5x + 2 = 0

4. Fourth degree equation - Quartic equation -

example: 2x

^{4}+ 3x

^{3}- 2x

^{2}+ 3x + 2 = 0

5. Fifth degree equation - Quintic equation -

example: 4x

^{5}+ 2x

^{4}- 3x

^{3}+ 2x

^{2}+ x - 4 = 0