trigonometry/ pre-calculus chapter p: prerequisites section p.4: solving equations algebraically
TRANSCRIPT
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.