Transposition of formulae

Question

Solve the equation f(x) = 0 for each of the following quadratic functions:

1.   f(x)=−x2 +22x−85
2.   f(x)=x2 −18x+81

Answer 1

1. -(5)^2 + 22(5) -85 = -25 + 110 -85 = 0 x=5 is one solution

-(17)^2 + 22(17) -85 = (17)(22-17) - 85 = (17)5) -85 = 85-85=0 x=17 is a 2nd solution

(9)^2 -18(9) + 81 = 81 - 162 + 81 = 0 x=9 is the only solution for the 2nd problem

solve both by completing the square, but the 2nd problem is already in a perfect square form

x^2 -18x + 81 = (x-9)^2 = 0. take the square root to get x-9 =0 or x=9

for the 1st, -x^2 +22x -85 = 0, it's easier to multiply by negative one to get

x^2 -22x + 85=0, move the 85 to the right side which changes its sign to

x^2 -22x = -85. Now "complete the square on the left side by taking half of 22 to get 11, then

square it, to get 121, then add it to both sides

x^2 -22x +121 = -85 + 121

(x-11)^2 = 36

now take the square root of both sides

x-11 = + or - 6

add 11 to both sides

x= 11+6 or 11-6

x=17 or 5