Algebra Basics

What will be covered:

• Order of Operations• Variables vs. Constants• The Quadratic Formula• Common Algebra Mistakes

What will be tested:Any basics that have been covered heavily in math courses up to and including Algebra I, all of which are prerequisites for this course.

Solve this:

)68( ?2

}4

])32(4[{)68( 3

32

3

Order of Operations

• Please• Excuse• My • Dear• Aunt• Sally

Parentheses - {[(a + b)]}Exponents - ab

Multiplication - a x b , a • bDivision - a/b , a ÷ b

Addition - a + bSubtraction - a - b

Order of Operations• Parenthesis

– First proceed through PEMDAS through the parenthesis ( )

– Next, follow PEMDAS through any brackets [ ]– Then, do PEDMAS through braces { }– Finally, do PEDMAS through chevrons < >– Don’t forget that parenthesis are implied around

the dividend and the divisor:

)38()26(

3826 33

Order of Operations

• Exponents– Exponents are concise ways of displaying that the

base is multiplied by itself:• 64 = 6 x 6 x 6 x 6

– A negative exponent means that you should invert the base and then multiply.

• 2-3 = ½ x ½ x ½ = 1/(23)– An exponent applies ONLY to the base it is

immediately attached to:• 5y2 = 5(y2) . . . NOT (5y)2

Order of Operations

• Exponents (con’t)– A fraction exponent means that you should take

the denominator root of the base:• 61/2 =• 6251/4 =

– When negatives and fractions are both present, you treat them separately.

• 2-1/4 = 2-1 x ¼ =

64 625

4

21

Order of Operations• Exponents (continued)

– Product of Powersam * an = am+n

– Power of a Power(am)n = amn

– Power of a Product(ab)m = am * bm

– Zero Exponenta0 = 1; unless a = 0, at which point a0 = 0

– Quotient of Powersam / an = am-n; a can not equal 0

– Powers of a Quotient(a / b)m = am / bm; b can not equal 0

532 4)444()44(44

632 3)33()33()33()3(

23

5

555555555

55

Order of Operations

• Simplify these:– 1. (x4)2 – 2. x3 + y3 – 3. 33 * 34 – 4. z8 / z11 – 5. (5x2y2)7 – 6. (x8 / xy)2

– 7. x-3/2

Product of Powersam * an = am+n

Power of a Power(am)n = amn

Power of a Product(ab)m = am * bm

Zero Exponenta0 = 1; unless a = 0, at which point a0 = 0

Quotient of Powersam / an = am-n; a can not equal 0

Powers of a Quotient(a / b)m = am / bm; b can not equal 0

Order of Operations

• Multiplication and Division– Since division is really just inverted multiplication,

we can do both steps at the same time, from left to right.

31

343

34

3

)34(

Order of Operations

• Addition and Subtraction– Since Subtraction is really just adding a negative

value, we can do both in the same step, from left to right.

5 - 2 = 5 + -2

6 - -4 = 6 + 4

NOW, Solve this:

?2

}4

])32(4[{)68( 3

32

3

Here we go:

?2

}4

])32(4[{)68( 3

32

3

?2

}4])1(4[{

)2168( 3

32

?8

}4]4[{

)224(

3

We can work on each term separately.

What did I do?

Now what did I do?(Science text will always skip steps, it’s up to you to figure out what they did!

Here we go:

?8

}4]4[{

)224(

3

?8

}464{

)224(

?8}16{)224(

4482)224(

Variables vs. ConstantsVariables are numbers that are dynamic and will change

as the other variables in the equation change to keep the statements true. For the very beginning of this class, variables will typically be indicated in italic font as x and y

Constants are numbers in an equation that do not change. They are typically coefficients and, for the beginning of this class, will be indicated by normal, lowercase letters from the beginning of the alphabet like a, b and c, or the first letter of the word they represent, like g for gravity.

The Quadratic Formula• A Quadratic Equation is any equation that can

be manipulated into the form:y = ax2 + bx + c

• Solutions to quadratic equations can be found using the formula:

aacbbx

242

*** Get the program QUADFORM on your calculator NOW!!!***

Common Algebra Mistakes:

• Combining factors:– Find the mistake:

– Correct:

• Solving Linear equations:– Find the mistake:

– Correct:

222 44 yyy

222 )4(4 yyy

902180 kk

901

1802

1801802180 kkk

Common Algebra Mistakes:

• Exponents:– Find the mistake:

– Correct:

• Exponents:– Find the mistake:

– Correct:

tt 13)3.1(10

tt )3.1(10)3.1(10

642

4242

Common Algebra Mistakes:

• Parenthesis:– Find the mistake:

– Correct:

• Simplifying Fractions:– Find the mistake:

– Correct:

2212122)12(122 hxhxxhx

hxhxxhx 212122)12(122

12

122

xxx

12

12

22

xx

xx

Common Algebra Mistakes:

• Simplifying Fractions:– Find the mistake:

– Correct:

• Simplifying Radicals:– Find the mistake:

– Correct:

hrhhhrh

22 2

hrhhrh

22 2

xxx

xxx 2

Common Algebra Mistakes:

• Solving Linear Expressions:– Find the mistake:

– Correct:

• Simplifying Radicals:– Find the mistake:

– Correct:

31863189 kkkk

5.118123189 kkkk

zyzy 22

2222 zyzy

Common Algebra Mistakes:

• Solving Linear Expressions:– Find the mistake:

– Correct:

• Solving Quadratic Functions:– Find the mistake:

bbxxb

xx 11

2

2

22

bx

xbx

x 22

22 11

44404

2222222 xyxyxyyx