#1**+3 **

This isn't as difficult as it seems...we just need to use some "creative" Algebra

4x^2 +9x - 6 = 0

In the form ax^2 + bc + c = 0

The sum of the roots, ( s + t ) = -b/ a = -9 / 4

The product of the roots, (st ) = c /a = -6/4 = -3/2

Note that we can write

s/t + t /s = [ s^2 + t^2 ] / st

And note that ( s + t)^2 = s^2 + 2st + t^2

So

(-/9/4)^2 = s^2 + 2(-3/2) + t^2

81/16 = -3 + [s^2 + t^2] ..... add 3 to both sides

81/16 + 3 = s^2 + t^2

81/16 + 48/16 = s^2 + t^2

129/16 = s^2 + t^2

So......

s/t + t/s = [ s^2 + t^2 ]/ st = [ 129/16] / [-3/2)] = [129/16] * [ -2/3] = -43 / 8

CPhill Jun 25, 2018

#2**+3 **

Here's the second one

2x^2 - 8x + 7 = 0

Let's write this in the form mx^2 + nx + q = 0 to avoid confusion with the roots

And we can write

1 / (2a) + 1 / (2b) = [a + b ] / [ 2ab ]

Using a similar idea as in the first one

The sum of the roots ( a + b) = -n/m = - (-8) / 2 = 8/2 = 4

The product of the roots ab = q/m = 7/2

So...

1/(2a) + 1/ (2b) =[ a + b ] / [2ab] = 4 / [ 2 (7/2) ] = 4 / 7

CPhill Jun 25, 2018