+207 Let a and b be the solutions to 7x^2 + x - 5 = -3x^2 - 9x - 4. Compute (a - 4)/(b - 4) + (b - 4)/(a - 4).
+129847 Simplify as
10x^2 + 10x - 1
ab = -1/10
2ab = -2/10 = -1/5
a + b = -1 square both sides
a^2 + 2ab + b^2 = 1
a^2 - 1/5 + b^2 = 1
a^2 + b^2 = 1 + 1/5 = 6/5
(a -4) / ( b -4) + ( b -4) / (a -4) = [ (a - 4)^2 + ( b -4)^2] ] ' [ (a -4) ( b-4) ] =
[ a^2 - 8a +16 + b^2 - 8b + 16 ] / [ ab - 4(a + b) +16 ] =
[ (a^2 + b^2) - 8(a + b) + 32 ] / [ ab - 4 (a + b) + 16 ] =
[ 6/5 - 8 (-1) + 32 ] [ -1/10 - 4 (-1) + 16 ] =
[ 6/5 + 40 ] / [ 20 - 1/10] = 20497 / 25
