Mathematics Higher Level Robert Joinson

©Sumbooks2002Higher Level82 Completing the Square1) In each of the following expressions determine the number which must be added (orsubtracted) to make a perfect square. Some have been done for you.x 2a) + 6xb) x 2 + 4xc) x 2 – 3xx 2 + 6x + c 2 = ( x + c) 2x 2 + 6x + c 2 = x 2 + 2cx + c 2So 6x=2cxThereforec = 3 and c 2 = 9i.e. 9 must be added to make a perfect squared) – 7xe) x 2 – xf) x 2 + 9xg) 3x 2 + 9xh) 4x 2 – 3xi) 7x 2 + 4x3x 2 + 9x + c 2 = ( 3x + c) 2SoTherefore ci.e.x 23x 2 + 9x + c 2 = 3x 2 + 2 3cx + c 29x = 2 3cx27-----49= --------- and c 2 = 81 ----- =2 31227-----4must be added to make a perfect squarej) 6x 2 + 2xk) 9x 2 – 3xl) 16x 2 + 5x2) Find the solution to these equations by first completing the square.a) – 8x – 20 = 0b) x 2 – 4x – 21 = 0c) x 2 + 11x + 18 = 0x 2d) x 2 + 5x + 6 = 0e) x 2 – x – 2 = 0f) x 2 + 3x – 4 = 0g) – 4x – 5 = 0h) x 2 + 3x – 28 = 0i) x 2 + 4x + 3 = 0x 2j) 2x 2 + 5x + 2 = 0k) 3x 2 – 4x – 4 = 0l) 2x 2 + 5x – 3 = 0m) 8x 2 + 2x – 1 = 0 n) 6x 2 + x – 2 = 0o) 15x 2 + 4x – 3 = 0