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Let $a$ and $b$ be the roots of the quadratic equation $2x^2 - 7x + 2 = -x^2 + 4x + 9.$ Find $\frac{1}{a+7}+\frac{1}{b+7}.$

bader Mar 8, 2024

CPhill · Answer 1 · 2024-03-08T06:41:55

Simplify as

3x^2 - 11x - 7 = 0

Product of roots ab = -7/3

Sum of roots = a + b = 11/3

1/ (a + 7) + 1/ (b + 7) = [ a + b + 14 ] / [ ab + 7 (a+b) + 49 ] =

[ 11/3 + 14 ] / [ -7/3 + 7(11/3) + 49 ] =

[ 53/3] / [ 70/3 + 49] =

[53/11] / [ 217/3] =

[53/3][ 3/217] =

53 / 217