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Name___Derivation of the Quadratic Formula Name___________________ Derivation of the Quadratic Formula General form of a quadratic equation: ax 2 + bx + c = 0, ∀a,b,c ∈ℜ, a ≠ 0 Algebraic Representations Directions b c x 2 + x + = 0, € a ≠ 0 Step 1: a a b c x 2 + x = − Step 2: € a a 2 2 2 b ⎛ b ⎞ c ⎛ b ⎞ Step 3: x + x + ⎜ ⎟ = − + ⎜ ⎟ a ⎝ 2a ⎠ a ⎝ 2a ⎠ 2 2 ⎛ b ⎞ c ⎛ b ⎞ Step 4: ⎜ x + ⎟ = − + ⎜ ⎟ ⎝ 2a ⎠ a ⎝ 2a ⎠ 2 2 ⎛ b ⎞ c b Step 5: ⎜ x + ⎟ = − + 2 ⎝ 2a ⎠ a 4a 2 2 ⎛ b ⎞ c 4a b Step 6: ⎜ x + ⎟ = − • + 2 ⎝ 2a ⎠ a 4a 4a 2 2 ⎛ b ⎞ b − 4ac Step 7: ⎜ x + ⎟ = 2 ⎝ 2a ⎠ 4a 2 ⎛ b ⎞ b2 − 4ac Step 8: ⎜ x + ⎟ = ± 2 ⎝ 2a ⎠ 4a 2 ⎛ b ⎞ b − 4ac Step 9: ⎜ x + ⎟ = ± 2 ⎝ 2a ⎠ 4a b b 2 − 4ac Step 10: x + = ± 2a 4a 2 b b2 − 4ac x + = ± Step 11: 2a 2a b b2 − 4ac x = − ± Step 12: 2a 2a − b ± b2 − 4ac x = Step 13: 2a MCC@WCCUSD 12/02/11 The steps below are in mixed order. Your task is to arrange them in the correct order. 1) Cut out the 13 steps below. 2) Use the algebraic expressions on the first page to help arrange the steps in the correct order. 3) Paste them in the correct order in the 2nd column. Simplify 4a2 on the right side of the equation. b Subtract from both sides of the equation. € 2a Divide the general form of a quadratic equation by a. € Factor the trinomial on the left side of the equation. Combine the fractions on the right side of the equation. a a Use the property = on the right side b b of the equation. Combine the fractions to obtain the Quadratic Formula. € c Subtract the constant from both sides of the a equation. 2 " b $ Multiply out on the right side of the €# 2a% equation. Take half of the coefficient of the linear term, square it, and add it to both sides of the equation.€ 2 " b $ Simplify x + on the left side of the # 2a% equation. c Multiply − by an equivalent form of one to a €obtain common denominators. Take the square root of both sides of the equation. € MCC@WCCUSD 12/02/11 Algebra 1 Name___________________ Derivation of the Quadratic Formula General form of a quadratic equation: ax 2 + bx + c = 0, ∀a,b,c ∈ℜ, a ≠ 0 Directions Algebraic Representations Divide the general form of a quadratic € Step 1: equation by a. c Subtract the constant from both sides of Step 2: a the equation. Take half of the coefficient of the linear term, square it, and add it to both sides of the Step 3: € equation. Factor the trinomial on the left side of the Step 4: equation. 2 " b $ Multiply out on the right side of the Step 5: # 2a% equation. c Multiply − by an equivalent form of one to Step 6: a obtain€ common denominators. Combine the fractions on the right side of the equation. Step 7: € Take the square root of both sides of the Step 8: equation. 2 " b $ Simplify x + on the left side of the Step 9: # 2a% equation. a a Use the property on the right side Step 10: = € b b of the equation. Simplify 4a2 on the right side of the Step 11: equation. € b €Subtract from both sides of the equation. Step 12: 2a Combine the fractions to obtain the Quadratic Formula. Step 13: € MCC@WCCUSD 12/02/11 The steps below are in mixed order. Your task is to arrange them in the correct order. 1) Cut out the 13 steps below. 2) Use the written directions on the first page to help arrange the steps in the correct order. 3) Paste them in the correct order in the 2nd column. 2 b b − 4ac 2 x = − ± ⎛ b ⎞ b − 4ac 2a 2a ⎜ x + ⎟ = ± 2 ⎝ 2a ⎠ 4a 2 2 ⎛ b ⎞ c ⎛ b ⎞ 2 2 ⎜ x + ⎟ = − + ⎜ ⎟ 2 b ⎛ b ⎞ c ⎛ b ⎞ 2a a 2a x + x + ⎜ ⎟ = − + ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ a ⎝ 2a ⎠ a ⎝ 2a ⎠ b b 2 − 4ac 2 2 x + = ± ⎛ b ⎞ b − 4ac 2a 2 ⎜ x + ⎟ = ± 4a 2a 4a 2 ⎝ ⎠ 2 ⎛ b ⎞ c b 2 2 x + = − + − b ± b − 4ac ⎜ ⎟ 2 x = ⎝ 2a ⎠ a 4a 2a 2 b c 2 x + x = − ⎛ b ⎞ c 4a b 2 a a ⎜ x + ⎟ = − • + 2 ⎝ 2a ⎠ a 4a 4a 2 ⎛ b ⎞ b 2 − 4ac 2 x + = b b − 4ac ⎜ ⎟ 2 x + = ± ⎝ 2a ⎠ 4a 2a 2a b c x 2 + x + = 0, a ≠ 0 a a € MCC@WCCUSD 12/02/11 Algebra 1 Name___________________ Derivation of the Quadratic Formula After today’s lesson, you should know the quadratic formula and be familiar with its proof by completing the square. You should also be able to solve quadratic equations by using the quadratic formula. (CA 19.0, 20.0) Algebraic Representations Directions b c Divide the general form of the quadratic x 2 + x + = 0, a ≠ 0 Step 1: a a equation by a. b c c x 2 + x = − Subtracted the constant from both sides of Step 2: € a a a the equation. 2 2 Take half of the coefficient of the linear term, 2 b ⎛ b ⎞ c ⎛ b ⎞ x + x + = − + squared it, and added it to both sides of the Step 3: ⎜ ⎟ ⎜ ⎟ € a ⎝ 2a ⎠ a ⎝ 2a ⎠ equation. 2 2 Factor the trinomial on the left side of the ⎛ b ⎞ c ⎛ b ⎞ Step 4: ⎜ x + ⎟ = − + ⎜ ⎟ equation. ⎝ 2a ⎠ a ⎝ 2a ⎠ 2 b 2 c b 2 " b $ ⎛ x ⎞ Multiply out on the right side of the Step 5: ⎜ + ⎟ = − + 2 ⎝ 2a ⎠ a 4a # 2a% equation. 2 2 c ⎛ b ⎞ c 4a b Multiply − by 1 to obtain common ⎜ x + ⎟ = − • + Step 6: 2a a 4a 2 a ⎝ ⎠ 4a denominator€ s. 2 2 Combine the fractions in the right side of the ⎛ b ⎞ b − 4ac Step 7: ⎜ x + ⎟ = 2 equation. ⎝ 2a ⎠ 4a € 2 Take the square root of both sides of the ⎛ b ⎞ b2 − 4ac equation. Step 8: ⎜ x + ⎟ = ± 2 ⎝ 2a ⎠ 4a 2 2 ⎛ b ⎞ b − 4ac " b $ ⎜ x + ⎟ = ± 2 Simplify x + on the left side of the Step 9: ⎝ 2a ⎠ 4a # 2a% equation. 2 a a b b − 4ac Use the property on the right side Step 10: x + = ± = 2a 4a 2 € b b of the equation. b b2 − 4ac Simplify 4a2 on the right side of the Step 11: x + = ± equation. 2a 2a € 2 b b b − 4ac Subtract from both sides of the equation. x = − ± € Step 12: 2a 2a 2a − b ± b2 − 4ac Combine the fractions to obtain the Quadratic x = Formula. Step 13: 2a € MCC@WCCUSD 12/02/11 .
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