Quadratic Equations

Quadratic Equations




where x represents a variable or an unknown, and a, b, and c are constants with a not equal to 0. (If a = 0, the equation is a linear equation.)

The constants a, b, and c are called, respectively, the quadratic coefficient, the linear coefficient and the constant or free term. Quadratic equations can be solved by factoring, completing the square, using the quadratic formula, and graphing. 



Here are some examples of a Quadratic Equation:
a) x²+2x-8=0    
b) x²+10x-18=0
c) 9x²+4x-13                     

Solutions
a)  x²+2x-8=0      *factor this equation
     (x+4)(x-2)       *find what must add to become 2x and find the factors of -8
     x=-4 , x=2        *subtract 4 to both sides (x+4) , add 2 to both sides (x-2)
b) x^{2 -}10x-18=0      *complete the square
   *determine a,b,c     (a=1,b=-10,c=-18)
    x^{2 -}10x    =18        *add 18 to both sides
    x^{2 -}10x+25=18+25       *find (b/2)^2 = (-5)^2 or 25
    x^{2 -}10x+25=43         *factor
   (x-5)(x-5)=43   or    (x-5)^2=43       *square both sides    
    x-5 = ±43^(1/2)     *add 5 to both sides   x = 5±43^(1/2)       


        

The Six Trigonometric Functions

The two basic trigonometric functions are: sine (which we have already studied), and cosine. By taking ratios and reciprocals of these functions, we obtain four other functionscalled tangent, secant, cosecant, and cotangent.

These are the core values to be used in the trigonometric functions. They are often called as SOHCAHTOA.

They can be used in the Pythagorean theorem. To determine the Opposite , Adjacent side as well as with the Hypotenuse.
So let's try to use them.

Notice that sin,cos,tan are just opposite with csc, sec and cot.

Lets say we had a right triangle that we wanted to find the secant. The 2 other sides were 
8cm and 4cm and the hypotenuse is 8.94cm. You would use the SOHCAHTOA in order 
to find the secant.


So that means Secant is...

Special Triangle

SPECIAL TRIANGLES

a² + b² = c²

1² + 1² = [sqrt (2)]²

a² + b² = c²

1² + [sqrt (3)]² = 2²

Logarithm

A logarithm is an exponent; Log_a x is the exponent to which the base a must be raised to obtain x.

The abbreviation log is used for the word logarithm. Read Log_a x as logarithm of x to the base a or the base a logarithm of x.

To remember the location of the base and the exponent in each form, refer to the following :

Logarithmic Form : y = log_a x               Exponential Form : x = a ^y

*where y is the exponent ;a is the base*

Example (conversion between logarithmic and exponential form):
1. 3² = 9 -------> 2 = log₃ 9
2. 5 = log₁₀₀ 100000 -------> 10⁵ = 1000000
3. 5² = 25 ---------> 2 = log₅ 25

Also:
log_a x = log x divided by log a (log x / log a)


Common and Natural Logarithm

Common Logarithm
     Base is always 10 (log)
log 10 or log₁₀ 10

Natural Logarithm
     Base e logarithm (ln)
ln e = 1 (where e is approximately equal to 2.718281828)
example:

e^x-1 = 4

ln e^x-1 = ln 4

(x-1) ln e = ln 4

x ln e - ln e = ln 4

x - 1 = ln 4

x = ln 4 + 1

x = 0.39

Properties of Logarithms

Product Rule:

log_b xy = log_b x + log_b y

Quotient Rule:

log_b x/y = log_b x – log_b y

Power Rule:

log_b x^r = r log_b x

Special Properties

log_b b^x = x

b log_b x = x

log_b 1 = 0

Linear Equation

Solving Linear Equations

Linear Equation:

-A mathematical expression that has an equal sign and linear expressions

Variables

-A number that you don`t know, often represented by `x` or `y` but any letter will do!

Variable(s) in linear expressions

-Cannot have exponents (or powers)

for example: x squared or x^2

-Cannot multiply or divide each other

for example: `x` times `y` or xy, `x` divided by `y` or x/y

-Cannot be found under a root sign or square root (sqrt)

for example: √x or the `square root x `; sqrt (x)

Linear Expressions:

-A mathematical statement that performs functions of addition, subtraction, multiplication, and division

These are examples of linear expressions

x+4 , 2x+4 , 2x+4y

These are not linear expressions:

x^2 (no exponents on variables)

3xy+5 (can`t multiply two variables)

2x/6y (can`t divide two variables)

√x (no square root sign or variables)

Solve these linear equations by clicking and dragging 
a number to the "other" side of the equal sign. 
Remember that you are "isolating" the unknown "X" to solve the problem. 
(More examples are provided below.)

Isolate x in order to solve the equation. Click on parts of the equation on the left side and drag them to the right side.

3x-8=7

More examples:

Linear equation, solving example #1

Find x if: 2x+4=10

1. Isolate `x` to one side of the equation by subtracting 4 to both sides: 
2x+4-4=10-4

2x=6

2. Divide both sides by 2: 
2x/2=6/2

x=3

3. Check your work with the original equation:
2x+4=10

2(3) + 4=10

6+4=10

Linear equation, solving example #2

Find x if: 3x-4=-10

1. Isolate `x` to one side of the equation by adding 4 to both sides: 
3x-4+4=-10+4

3x=-6

2. Divide both sides by 3: 
3x/3=-6/3

 x=-2

3. Check your work with the original equation: 
3(-2)-4=-10

-6-4=-10

Linear equation, solving example #3:

Find x if: 4x-4y=8

(using more than one variables)

1. First step is to isolate `x` to one side of the equation by adding 4y to both sides: 
4x-4y+4y=8+4y

 4x=8+4y

2. Second step is to divide both sides by 4: 
4x/4 = (8+4y)/4

 x=2+y

3. Check your work with the original equation: 
4(2+y) - 4y=8

8+4y-4y=8

8=8

Linear equation, solving example #4:

Find x if x+3^2 =12

Note: since the square is on the number and not on the variable, the expression qualifies as a linear expression

1. First step is to square the number: 
x+3^2=12

x+9=12

2. Second step is to subtract both sides by 
9: x+9-9=12-9

 x=3

3. Check your work with the original equation:
3+3^2=12

12=12

Inequalities

Basic Rules

Symbol	Words	Example
>	greater than	x + 3 > 2
<	less than	7x < 28
≥	greater than or equal to	5 ≥ x - 1
≤	less than or equal to	2y + 1 ≤ 7

Here is an example: Consider the inequality

When we substitute 8 for x, the inequality becomes 8-2 > 5. Thus, x=8 is a solution of the inequality. On the other hand, substituting -2 for x yields the false statement (-2)-2 > 5. Thus x = -2 is NOT a solution of the inequality. Inequalities usually have many solutions.

As in the case of solving equations, there are certain manipulations of the inequality which do not change the solutions. Here is a list of "permissible'' manipulations:

Rule 1. Adding/subtracting the same number on both sides.


Example: The inequality x-2>5 has the same solutions as the inequality x > 7. (The second inequality was obtained from the first one by adding 2 on both sides.)

Rule 2. Switching sides and changing the orientation of the inequality sign.
Example: The inequality 5-x> 4 has the same solutions as the inequality 4 < 5 - x. (We have switched sides and turned the ``>'' into a ``<'').

Last, but not least, the operation which is at the source of all the trouble with inequalities:

Rule 3a. Multiplying/dividing by the same POSITIVE number on both sides.

Rule 3b. Multiplying/dividing by the same NEGATIVE number on both sides AND changing the orientation of the inequality sign.


Examples: This sounds harmless enough. The inequality  has the same solutions as the inequality  . (We divided by +2 on both sides).
The inequality -2x > 4 has the same solutions as the inequality x< -2. (We divided by (-2) on both sides and switched ">'' to "<''.)
But Rule 3 prohibits fancier moves: The inequality  DOES NOT have the same solutions as the inequality x > 1. (We were planning on dividing both sides by x, but we can't, because we do not know at this point whether x will be positive or negative!) In fact, it is easy to check that x = -2 solves the first inequality, but does not solve the second inequality.

Only ``easy'' inequalities are solved using these three rules; most inequalities are solved by using different techniques.

Examples:

1. solve 2x - 5 <12


 solution:

 2x -5 < 12

(2x - 5) + 5 < 12 + 5

2x < 17

(1/2)2x < (1/2)17

x < 17/2

                           The solution set of the inequality is {x.:.x.<.17/2} which is read as "the set of all x such that x is less than 17/2"

2. Solve 13 - 3x >= 10

 Solution:

 13- 3x >= 10
 -3x >= 10 - 13


-3x >= 1

 (-1/3)(-3x) <= (-1/3)(-3) (Recall - rule 5)

x <= 1

            The solution set is { x : x <=1 }

3. Solve 14(x-2) <=132 - 281x

Solution:

14(x-2) <= 132 - 281x

14x - 28 <= 132 - 281x

14x <= 160 - 281x

14x <= 160 - 281x

295x <= 160

x <= 160/2995

x <= 32/59

The solution set is { x.:.x.<= 32/59}