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Completing the square

Completing the square. Solving quadratic equations. Express the followings in completed square form and hence solve the equations. 1. x 2 + 4x – 12 = 0. 2. x 2 + 6x + 4 = 0. = (x + 2) 2 – 2 2 – 12 = 0. = (x + 3) 2 – 3 2 + 4 = 0. (x + 2) 2 – 16 = 0. (x + 2) 2 – 5 = 0.

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Completing the square

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  1. Completing the square Solving quadratic equations Express the followings in completed square form and hence solve the equations 1. x2 + 4x – 12 = 0 2. x2 + 6x + 4 = 0 = (x + 2)2 – 22 – 12 = 0 = (x + 3)2 – 32 + 4 = 0 (x + 2)2 – 16 = 0 (x + 2)2 – 5 = 0 (x + 2)2 = 16 (x + 3)2 = 5 x + 2 = 16 x + 3 = 5 x + 2 =  4 x = - 3  5 x = - 2  4 x = - 3 - 5 or - 3 + 5 x = -6 or x = 2

  2. y x (-1, 0) (5, 0) Sketching graph Express x2 - 4x -5 in the form (x + p)2 + q, hence: i) find the minimum value of the expression y =x2 - 4x - 5 . ii) solve the equation x2 - 4x - 5 = 0 iii) sketch the graph of the functiony =x2 - 4x - 5 Completed square form (x – 2)2 – 9 x2 – 4x – 5 = (x – 2)2 – 4 - 5 = Solving: x2 – 4x – 5 = 0 x2 – 4x – 5 = (x – 2)2 – 9 = 0 (x – 2)2 = 9 x – 2 = 9 x – 2 =  3 x = 2  3 Vertex (2, -9) x = -1 or x = 5 The curve is symmetrical about x = 2

  3. y x Sketching graph Write 1 + 4x - x2in completed square form, hence solve 1 + 4x – x2 = 0 and sketch the graph of y = 1 + 4x – x2. Completed square form 1 + 4x – x2 = - [ x2 – 4x ] +1 -[ x2 – 4x ] + 1 = - [ (x – 2)2 – 4 ] + 1 = - (x – 2)2 + 4 + 1 = - (x – 2)2 + 5 - (x – 2)2 + 5 = 0 Vertex (2, 5) - (x – 2)2 = - 5 (x – 2)2 = 5 x – 2 = 5 (2 - 5) (2 + 5) x = 2 5 x = 2 -5 or x = 2 + 5 The curve is symmetrical about x = 2

  4. y x (1 -(1/3), 0) (1 +(1/3), 0) Sketching graph Write -3x2 + 6x - 2 in completed square form, hence solve -3x2 + 6x – 2 and sketch the graph of y = -3x2 + 6x – 2. Completed square form -3[x2 - 2x ]– 2= -3[(x - 1)2 - 1 ]- 2 = -3(x - 1)2 + 3 -2 = -3(x - 1)2 + 1 -3(x - 1)2 + 1 = 0 -3(x - 1)2 = - 1 Vertex ( 1, 1 ) The curve is symmetrical about x = 1

  5. More examples Complete the square for each of the following quadratic functions and solve f(x) = 0 (a) x2 + x – ½ = (x + ½ )2 – ¼ – ½ = (x + ½ )2 – ¾ (x + ½ )2 – ¾ = 0 (x + ½ )2 = ¾ x + ½ = ¾ x = -½ ¾ 2[(x + 1 )2 – 1 ]+ 3 (c) 3 + 4x – 2x2 = -2 [x2 + 2 x ]+ 3 = = 2(x + 1 )2 + 1 = 2(x + 1 )2 – 2+ 3 2(x + 1 )2 + 1 = 0 2(x + 1 )2 = - 1 (x + 1 )2 = - ½  No solution

  6. The function f is defined for all x by f(x) = x2 + 3x– 5. a) Express f(x) in the form (x + P)2 + Q.  Complete the square

  7. Tip: Simplify the surd where b) Hence, or otherwise, solve the equation f(x) = 0, giving your answers in surd form.  Solve the equation f(x) = 0 by making x the subject, using the completed square format Tip: You could have used the quadratic formula on x2 + 3x– 5 = 0.

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