How many gallons of a 50% antifreeze solution must be mixed with 80 gallons of 30% antifreeze to get a mixture that is 40% antifreeze? Use the six-step method. You need ___ gallons.

I don't know what the "six-step method" is, but I do know how to solve the problem:

Let x = number of gallons of 50% solution

amount of antifreeze in 50% solution = 0.50x
amount of antifreeze in 10% solution = 0.10(80) = 8
amount of 40% solution = 0.40(x+80) = 0.40x + 32

Then, 0.50x + 8 = 0.40x + 32

0.10x = 24

x = 240 gal

The square root of √512k2 is 2(2/9) *√k2