Find all values of x that satisfy the equation\( \frac {12x}{x^2 + 8} = 2. \)\(Let $s$ and $t$ be the solutions of the quadratic $4x^2 + 9x - 6 = 0.$ Find $$\frac st + \frac ts.$$\)
\(Let $a$ and $b$ be the solutions of the quadratic equation $2x^2 - 8x + 7 = 0$. Find \[\frac{1}{2a} + \frac{1}{2b}.\]\)
+37157 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
+130079 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
