algebra basics
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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