Gabe amodeo, a nuclear physicist, needs 30 liters of a 60% acid solution. he currently has a 40% solution and a 70% solution. how many liters of each does he need to make the needed 30 liters of 60% acid solution?

x = liters of 40% solution

y = liters of 70% solution

Gabe needs 30 liters, so

x + y = 30

Each liter of 40% solution contributes 0.4x liter of acid; each liter of 70% solution contributes 0.7y liter of acid. Gabe needs a solution with concentration 60%, which from 30 liters amounts to a net volume of 0.6*30 = 18 liters of acid. So

0.4x + 0.7y = 18

Solve the system however you like. By substitution, we have

y = 30 - x

0.4x + 0.7(30 - x) = 18

0.4x + 21 - 0.7x = 18

3 = 0.3x

==> x = 10 ==> y = 20

"standard form:

3x2 + 14x − 10 = 0

solutions based on quadratic formula:

x1   = −14 − √ 142 − 4×3×(−10) /   2×3   = −14 − 2 × √ 79 /   6   ≈ −5.296

x2   = −14 + √ 142 − 4×3×(−10) /   2×3   = −14 + 2 × √ 79 /   6   ≈ 0.629