Download By Cameron Hilker Grade 11 Toolkit Exponents, Radicals, Quadratic

By Cameron Hilker
Grade 11 Toolkit
Exponents, Radicals, Quadratic formula, Factoring and RaVEs
Review of Linear Equations
Linear Equations
- equations with a degree of one that gives only one potential answer
- get the letters on one side of the equal sign and the numbers on the other
Examples: Solve for the given equation:
b) (3x-2/2+2x/3) =2x
6(3x-2/2+2x/3) =6(2x)
9x-6+4x=12x
9x+4x-6x=6
x=6
1. a) 2x+4=14
2x=14-4
2x=10
2 2
x=5
Remember Exponent Rules
If (xn)(xm)=xn+m then (x5)(x3)=x8 (same base)
If (xn)/(xm)=xn- m then x9/x2= x7 (same base)
If (xn)m=xnm then (x3)4= x12
If (xy)n=xnyn then (xy)7=x7y7
If( X/Y)n= xn/yn then (x2/y)7= x14/y7
If x0=1 then 7(x0) =7
If x-n=1/xn then 3-4=1/81 for fractions flip it: (2/3)-2=(3/2)2
Radicals
A radical means a root of a number so  3 4
Radicals must also be written in simplified form
so 700 = 107. These are done by transitions between entire radical (entire number under radical) and
mixed radical which is a number multiplied by the radical.
Example: 1a) show 32 as a mixed radical
32
16x2
42
b) show 75 as a entire radical
75
49x5
245
To multiply and divide radicals you do so with the one outside the radicals and so with the ones inside
the radicals. To add and subtract you need like radicals
Example: 2. Simplify
a) 25+45
= 25+9x5
= 25+35
= 55
b) 23 x 74
= 1412
By Cameron Hilker
Solving Exponential Equations
Solve for x: 2x=8
2x=x3
x=3
-adjust the left 8 right side so they are written in the same base
-when the bases are the same cancel them out and solve the linear equation
Examples: Solve
1. a) 51-3x=1
53-3x=50
3-3x=0
-3x=-3
-3 -3
x=1
b)35-x=1/3
35-x=3-1
5-x=-1
-x=-1-5
-x=-6
-1 -1
x=6
c)4(8x-3)=32
22(23(x-3)) =25
2+3(x-3) =5
2+3x-9=5
3x= 5+9-2
3x=12
3 3
x=4
Rational Exponents
Recall that rational is a number that can be written in the
form m/n.
If (51/2) (51/2) = 51/2+1/2= 51or 5
(50.5) (50.5)
Then what does the 1/2 exponent represent?
( 5)( 5) = 5
In general, (nx)m =xm/n or nxm=xm/n
Examples: 1. a) Express 483 as a single power
4
83
=83/4
b) Express (4/9)-3/2 as radical then evaluate
(4/9)-3/2
=4/9-3
=9/43
= (3/2)3
= 27/8
Finding the Roots of a Quadratic by Quadratic Formula
The quadratic formula is: x=-b±b2-4ac
2a
This formula will find numbers that can’t be factored out (decimals) and non-real numbers.
Example: Use the Quadratic formula to solve
1. x2-4x-12=0
x=-b±b2-4ac
2a
x= 4± (-4)2-4(1) (-12)
2(1)
By Cameron Hilker
x= 4± 16+48
2
x= 4± 64
2
x= 4 + 64
2
x= 4+8
2
x= 12
2
x=6 and -2
x= 4 - 64
2
x= 4-8
2
x= -4
2
Common Factoring
Factoring is the reverse of simplifying
To Common Factor:



Divide the common devisor from every coefficients
Remove the common variable that is in every term
Write the common factor in front and the remainder in brackets
Examples:
1. Expand
a) 7(2x+3y)
= 14x+21y
b) z(x-y)
= zx-zy
2. Factor
a) 14x+21y (÷7)
=7(2x+3y)
b) zx-zy
= z(x-y)
Difference of Square Factoring and Perfect Squares
This happens when after common factoring you notice that the first and last term are perfect squares.
Examples: 1. Factor
a) y2-144
= (y-12) (y+12)
b) 48y2-147
= 3(16y2-49)
= 3(4y-7) (4y+7)
c) 81b2+180b+100
= (9b+10) (9b+10)
Trinomial Factoring A=1
Remember FOIL:
(x+7) (x+8)
= x2 +8x+7x+56
= x2+15x+56
To change the trinomial into two binomials you will need to find two numbers
that get the sum of the middle term and the sum of the third term.
This must be in ax2+bx+c form to work
Example:
1. a) x2+14x+49
b) x2+5x+6
c) 4y2+16y+16
By Cameron Hilker
Sum
product
= (x+7) (x+7)
or (x+7)2
Sum product
= 4(y2+4+4)
sum product
= (x+3) (x+2)
= 4(y+2)(y+2) or 4(y+2)2
Trinomial Factoring A ? 1
ax2+bx+c when a doesn’t equal one it’s called complex factoring meaning you have a number for a that
can’t be factored out
Steps:
1. List the possible terms that could fit into the first and third term
2. Multiply diagonally for two answers and if their sum adds up to the middle term keep going if not try
step 1 again.
3. Write out the two binomials in brackets.
Examples:
1.Factor
a) 16y2-12y-40
4y
5 = 20
4y
-8 = -32
(4y+5) (4y-8)
b) 42y2+144y+54
2(21y2+72y+27)
7y
3=9
3y
9 = 63
2 (7y+3) (3y+9)
Finding the roots of a Quadratic by Factoring x2
Another way to solve is by factoring when the equation is in the form ax2+bx+c= 0. You must have it
equal to zero that means x will have two answers.
Example: Solve by Factoring
x2+9x+18
(x+3)(x+6)
x+3=0 x+6=0
x=-3
x=-6
x=-3 and-6
When 1 doesn’t equal zero you just factor it out the same way as usual and solve as usual
Example: Solve by Factoring
5x2+23x+12
5x
3 3x
x
4 20x
(5x+3)(x+4)
x=-3/5x, -4
By Cameron Hilker
Rational Variable Expressions (RaVEs)
Rules:
-
For monomial numerators and denominators: multiply across tops and bottoms then reduce.
For polynomial numerators and denominators: factor everything possible. State when
undefined, reduce and/or cancel (like binomials top with bottom)
Examples: Simplify
1 a) 14x3y2 x 39x4y3
35xy 2x3y4
546x7y5
70x4y5
39x3
5
x≠0 y≠0
b) 5(2x-1)(x+6) _
20(x+6)(2x+1) (restrictions found here)
1(2x-1)
4(2x+1)
x≠-6 & -1/2
c) x2+10x+21
x3+7x2
(x+3)(x+7) (Factor)
x2(x+7) (Common Factor)
x+3
x2
x≠0 & -7
Multiplying and Dividing RaVEs
If you are multiplying or dividing RaVEs that contain binomials and trinomials YOU MUST FACTOR FIRST!
Then you must find when undefined BEFORE cancelling.
Examples: Simplify and state restrictions
1 a) x2-25 x x2-7x+12
2x2-6x x2+x-20
(x+5)(x-5) x (x-3)(x-4)
2x(x-3)
(x+5)(x-4)
x-5
2x
x≠0, 3, -5, 4
b) 2x2 -5x-3_ ÷ 4x2+4x+1
2x2-11x+15 4x2-8x+5
(2x+1)(x-3) ÷ (2x+1)(2x+1)
(2x-5)(x-3) (2x+1)(2x-5)
(2x+1)(x-3) x (2x+1)(2x-5)
(2x-5)(x-3) (2x+1)(2x+1) (restrictions found here)
1
x≠ -1/2 & 5/2 & 3
Adding and Subtracting RaVEs
Reduce within individual rationales first (must be factored), find common denominator then combine
and simplify numerator and factor if possible
Examples: 1 a) 7x-2y – 12x-5y
xy2
x2y
7x2-2xy – 12xy-5y2
x2y2
x2y2
2
7x -2xy -12xy-5y2
7x2-14xy-5y2
b) x2-4x+4 – 16x2-8x+1
3x2-5x-2 4x2-9x+2
(x-2)(x-2) – (4x-1)(4x-1)
(3x=1)(x-2) (4x-1)(x-2)
(x-2)(x-2) – (4x-1)(3x+1)
(3x-1)(x-2) (3x-1)(x-2)
x2-2x-2x+4-12x2-4x+3x+1
(3x-1)(x-2)
2
-11x -5x+5
(3x-1)(x-2)