Find all values of x that satisfy the equation12xx2+8=2.Let$s$and$t$bethesolutionsofthequadratic$4x2+9x−6=0.$Find$$st+ts.$$
Let$a$and$b$bethesolutionsofthequadraticequation$2x2−8x+7=0$.Find\[12a+12b.\]
+37165 Multiply both sides by x^2 +8 to get
12x = 2x^2 + 16 re-arrange
0 = 2x^2 -12x + 16 Divide both sides by two
0=x^2 - 6x + 8 Factor
(x-4)(x-2) = 0
For this to be true either x-4 = 0 or x-2 = 0 so x= 4 or 2
+130466 4x^2 + 9x - 6 = 0
s/t + t/s = [ s^2 + t^2 ] [ st ]
Using the quadratic function to solve
x = ( -9 ± √ [ (9^2 - 4* 4 * - 6) ] ) / 8
x = ( -9 ± √ [ 177] )/ 8
Call s = ( -9 + √ [ 177] )/ 8 Call t = ( -9 - √ [ 177] )/ 8
s^2 = [ 81 - 18√ 177 + 177] / 64 = [258 - 18√177 ] / 64
t^2 = [ 81 + 18√177 + 177 ] / 64 = [ 258+ 18√177] / 64
s^2 + t^2 = 516 / 64 = 129/16
And st will = c / a = -6/4 = -3/2
So
[ s^2 + t^2 ] / st = 129/16 * -2 / 3 = - 43 / 8
