1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 MULTIPLICATION CHART EXPANDING a(b + c) = ab + ac (a – b)2 = a2 – 2ab + b2 (a + b)2 = a2 + 2ab + b2 (a + b) (c + d) = ac + ad + bc + bd (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a – b)3 = a3 – 3a2b + 3ab2 – b3 FACTORING a2 – b2 = (a + b) (a – b) a3b – ab = ab (a + 1) (a – 1) a2 + 2ab + b2 = (a + b)2 a3 + b3 = (a + b) (a2 – ab + b2) a2 – 2ab + b2 = (a – b)2 a3 – b3 = (a – b) (a2 + ab + b2) ROOTS OF A QUADRATIC The solution for a quadratic equation in the form of ax2 + bx + c = 0 can be found by using the quadratic formula: Algebra Properties of Addition and Multiplication Commutative property of addition a + b = b + a Commutative property of multiplication ab = ba Associative property of addition a + (b + c) = (a + b) + c Associative property of multiplication a(bc) = (ab)c Distributive property of multiplication over addition a(b + c) = ab + ac Distributive property of multiplication over subtraction a(b – c) = ab – ac x-axis y-axis quadrant -5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 (4,3) (4,-3) (-4,-3) (-4,3) THE COORDINATE PLANE LOGARITHMS log a x = n ↔ x = an (log to the base a) loga (xy) = loga x + loga y loga ( x ) = loga x – loga y loga xp = p log a x loga ax = x alog a x = x loga x = logb x logb a y Common log: A log that is written without a base: log x = n, the base is 10: log10 x = n. All loga rules apply. Natural log: A log that is written ln x = n, where the base is e: loge x = n. All loga rules apply. LAW OF EXPONENTS x = –b + b2 – 4ac =+++++++++++++ 2a If a, b e R, a, b 0 and p, q e Q, then: a– p = 1 (a 0) ap 1 2 3 4 5 6 7 8 apaq = ap+q ap aq = ap–q a p= ap (b 0) (b) bp (ap)q = apq (ab)p = apbp ap = q ap q =+++ a0 = 1 (a 0) (a 0) R–10 RESOURCE PAGES
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