+577 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).
+129840 Simplify as
10x^2 + 10x - 1 = 0
ab = -1/10
a + b = -10/10 = -1 square both sides
a^2 + b^2 + 2ab = 1
a^2 + b^2 = 1 - 2(-1/10) = 1+ 2/10 =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 + 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] =
412 / 199
