algebra basics. what will be covered: order of operations variables vs. constants the quadratic...
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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 • b
Division - a/b , a ÷ b
Addition - a + b
Subtraction - 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(
38
26 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
2
1
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
5555
55555
5
5
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.
3
1
3
43
3
4
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
}4
64{
)224(
?8
}16{)224(
4482)224(
Variables vs. Constants
Variables 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:
a
acbbx
2
42
*** 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
90
1
180
2
180
1802180 k
kk
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
1
2
1
22
xx
x
1
2
1
222
x
x
x
x
Common Algebra Mistakes:
• Simplifying Fractions:– Find the mistake:
– Correct:
• Simplifying Radicals:– Find the mistake:
– Correct:
hrhh
hrh
2
2 2
hrh
hrh
2
2 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:
bbx
xb
xx 1
12
2
22
bx
xbx
x 2
22
2 11
44404
2222222 x
yx
yxyyx