Trigonometry/Pre-Calculus

Chapter P: Prerequisites

Section P.4: Solving Equations Algebraically

Equations and Solutions to Equations

An equation is a statement that two algebraic expressions are equal.

To solve an equation in x means to find all values of x for which the equation is true.

Isolate x.(i.e. Get x by itself.) When solving an equation involving fractions,

clear the fractions by multiplying by the Least Common Denominator (LCD)

Clearing Fractions

Solve : 3

25x 2

159

100

(100)3

25x (100)2 (100)

159

100 LCD100

4 3x (100)2 1159

12x 200 159

12x 41 x 4112

4 1

Intercepts of a Graph

The x-intercept is the point at which the graph crosses the x-axis. (a, 0) Let y = 0, and solve for x.

The y-intercept is the point at which the graph crosses the y-axis. (0, b) Let x = 0, and solve for y.

Intercepts and Solutions

In order to solve an equation, set y = 0, which is the same process for finding the x-intercept.

So, the x-intercepts of the graph ARE the solutions to the equation.

Points of Intersection of Two Graphs

An ordered pair that is a solution of two different equations is a point of intersection.

To solve for the point of intersection of two equations, solve one equation for one variable and substitute that expression into the other equation.

Set them equal to one another if solved for y.

Example

Solve : 3x y 5

y 2x 15

Solve this equation for y : 3x y 5

3x 3x

y 3x 5

Now, substitute and solve : 3x 5 2x 15

Extraneous Solutions???

When multiplying or dividing by a variable expression, it is possible to introduce an extraneous solution.

This is a solution that DOES NOT satisfy the original equation. Ignore

If a solution makes a denominator zero, is it

extraneous.

Polynomial Equations

Polynomial equations are classified by their degree (the greatest power of the variable): First degree - linear equation

Second degree - quadratic equation

Third degree - cubic equation

6x 2 4

3x 2 5x 7

x 3 x 0

Solving Quadratics

The quadratic formula can ALWAYS be used to solve a quadratic equation.

First, get the equation in the form:

ax 2 bx c 0

x b b2 4ac

2a

Solving Polynomial Equations by Factoring

Get all terms on one side of the equation set equal to zero.

Factor out the G.C.F. FIRST Now factor the resulting polynomial using the

various rules. Difference of perfect squares Sum or difference of perfect cubes “Reverse FOILing” of trinomials

Factoring Binomials

Difference of PERFECT SQUARES :

ex. x 2 16

x 4 x 4

Difference of PERFECT CUBES:

ex. x 3 64

First, rewrite : x 3 43

Pattern : x 4 x 2 4x 16

Opposite sign

Factoring Binomials (cont.)

Sum of PERFECT CUBES :

ex. x 3 64

First, rewrite : x 3 43

Pattern : x 4 x 2 4x 16

Opposite sign

Trinomials

Reverse FOILing

ex. x 2 5x 6

x 3 x 2

ex. x 2 5x 6

x 3 x 2

If the second sign is positive, signs are the same

and both the same as the first sign.

Trinomials (cont.)

Reverse FOILing

ex. x 2 x 6

x 3 x 2

ex. x 2 x 6

x 3 x 2

If the second sign is negative, signs are opposite.

The Berger Method

Used when coefficient of x 2 term is not 1.

1. Multiply the coefficient to the constant term.

2. Factor the trinomial as you would.

3. Divide back by the coefficient.

4. Simplify each fraction.

5. If denominator remains, rewrite with that as

the coefficient of the x term.