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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 = 107. 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 42 b) show 75 as a entire radical 75 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) 25+45 = 25+9x5 = 25+35 = 55 b) 23 x 74 = 1412 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, (nx)m =xm/n or nxm=xm/n Examples: 1. a) Express 483 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)