Question

v/v+11 + 1/v+19=1

Answer 1

Multiply to get a common denominator:

The common denominator must be (v+11)(v+19). Therefore, multiply each part by the missing factor to reach that denominator.

v/(v+11)*(v+19)/(v+19) + 1/(v+19)*(v+11)/(v+11) = (v+11)(v+19)/((v+11)(v+19))

Now we have v(v+19)/((v+11)(v+19)) + (v+11)/((v+11)(v+19)) = (v+11)(v+19)/((v+11)(v+19))

We can eliminate the denominator now. The equation becomes
v(v+19)+v+11=1
v^2 + 19v + v + 11 = 1
v^2 + 20v + 10 = 0

Use the quadratic equation to find v.
v = (-20 +/- sqrt(400-4*10*1))/2
v = (-20 +/- sqrt(360))/2
v = (-20 +/- 6*sqrt(10))/2
v = -10 +/- 3*sqrt(10)