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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 x2 + 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 = a2 + 2ab + b2 (a - b)2 = a2 - 2ab + b2 What is really important, is the relationship between ±2b and b2. 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.**

For ax2 + 2bx + b2 = 0 2b = (+8) This could have been solved faster by factoring the original quadratic equation. x2 + 8x = 0 x2 + 8x = 0 To complete the square we need to add b2 = +16 to both sides. x2 + 8x + 16 = 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! x2 + 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.

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**Simple Example 2 Example 3 x2 + 8x + 12 = 0 x2 - 5x - 3 = 0**

Rearrange to x2 + bx = c x2 - 5x - 3 = 0 Rearrange to x2 + bx = c x2 + 8x = -12 For ax2 + 2bx + b2 = 0 2b = (+8) x2 - 5x = 3 For ax2 - 2bx + b2 = 0 2b = (-5) To complete the square we need to add b2 = +16 to both sides. To complete the square we need to add b2 = +25/4 to both sides. x2 + 8x + 16 = x2 + 8x + 16 = 4 Factorise LHS, simplify RHS. Factorise LHS, simplify RHS. (x + 4)2 = 4 Take √ of both sides. Take √ of both sides. x + 4 = ±2 Solve for x. Solve for x. x = -4±2 i.e. x = or x = x = -6 or x = -2 The solution is x = -8 and x = 0. 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) x2 + 6x = 0 b) x2 - 7x = 0 c) x2 - 10x = -9 d) x2 - 6x + 3 = 0 e) x2 + 9x - 5 = 0 f) x2 - 3x - 7 = 0

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**Harder Examples: Solve by complete the square method.**

Divide by 3 and rearrange to x2 + bx = c a) Divide by 2 and rearrange to x2 + bx = c b) For ax2 + 2bx + b2 = 0 2b = For ax2 + 2bx + b2 = 0 2b = (+4) To complete the square we need to add b2 = to both sides. To complete the square we need to add b2 = +4 to both sides. You need to take CARE!. Factorise LHS. Factorise LHS. Take √ of both sides. Solve for x. Take √ of both sides. Solve for x. For ax2 + 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) 2x2 - 5x - 4 = 0 b) 3x2 + 3x - 5 = 0 c) 2x2 - 7x + 1 = 0 d) 5x2 + 4x - 2 = 0 e) 3x2 - 10x + 2 = 0

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