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Revision Quadratic Equation Solving Quadratic Equation by Complete the Square Method. By I Porter

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Introduction Factoring to solve quadratic equations is generally quite fast and easy, but many quadratics cannot be solved by factorisation. For example, the quadratic x 2 + 5x - 1 = 0, appears to be simple, but cannot be factorised. For these, we must use the COMPLETE THE SQUARE method or the Quadratic Formula. The Complete the Square method is based on the following two expansion: (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 What is really important, is the relationship between ±2b and b 2. This is an important step in the complete the square method of solving quadratic equations.

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Simple Example 1: Solve the following quadratic equations by complete the square method. x 2 + 8x = 0 For ax 2 + 2bx + b 2 = 0 2b = (+8) To complete the square we need to add b 2 = +16 to both sides. x 2 + 8x + 16 = x 2 + 8x + 16 = 16 Factorise LHS, simplify RHS. (x + 4) 2 = 16 Take √ of both sides. x + 4 = ±4 Solve for x. x = -4±4 i.e. x = or x = x = -8 or x = 0 The solution is x = -8 and x = 0. This could have been solved faster by factoring the original quadratic equation. x 2 + 8x = 0 x (x + 8) = 0 therefore x = 0 or x + 8 = 0 x = 0 or x = -8 We obtain the same result, but a lot faster!

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Simple Example 2 x 2 + 8x + 12 = 0 For ax 2 + 2bx + b 2 = 0 2b = (+8) To complete the square we need to add b 2 = +16 to both sides. x 2 + 8x + 16 = x 2 + 8x + 16 = 4 Factorise LHS, simplify RHS. (x + 4) 2 = 4 Take √ of both sides. x + 4 = ±2 Solve for x. x = -4±2 x = -6 or x = -2 x 2 + 8x = -12 Rearrange to x 2 + bx = c i.e. x = or x = The solution is x = -8 and x = 0. Example 3 x 2 - 5x - 3 = 0 Rearrange to x 2 + bx = c x 2 - 5x = 3 For ax 2 - 2bx + b 2 = 0 2b = (-5) To complete the square we need to add b 2 = + 25 / 4 to both sides. Factorise LHS, simplify RHS. Take √ of both sides. Solve for x. The solutions should be written in exact form as above.

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Exercise 1 Solve the following quadratic equations by complete the square method. a) x 2 + 6x = 0 b) x 2 - 7x = 0 c) x x = -9 d) x 2 - 6x + 3 = 0 e) x 2 + 9x - 5 = 0 f) x 2 - 3x - 7 = 0

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Harder Examples: Solve by complete the square method. a) Divide by 2 and rearrange to x 2 + bx = c For ax 2 + 2bx + b 2 = 0 2b = (+4) Factorise LHS. Take √ of both sides. Solve for x. To complete the square we need to add b 2 = +4 to both sides. b) Divide by 3 and rearrange to x 2 + bx = c For ax 2 + 2bx + b 2 = 0 2b = Take √ of both sides. Solve for x. Factorise LHS. To complete the square we need to add b 2 = to both sides. You need to take CARE!. For ax 2 + bx + c = 0, if a ≠1 or 0, step 1 is to divide every term by ‘a’.

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Exercise 2: Solve by complete the square method. a) 2x 2 - 5x - 4 = 0 b) 3x 2 + 3x - 5 = 0 c) 2x 2 - 7x + 1 = 0 d) 5x 2 + 4x - 2 = 0 e) 3x x + 2 = 0

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