ANSWER:
Jack = 9.12 hours
Lucy = 7.12 hours
STEP-BY-STEP EXPLANATION:
Given:
Jack + Lucy = 4 hours
Jack = x hours
Lucy = (x - 2) hours
From the statement we can propose the following equation:
[tex]\frac{1}{x}+\frac{1}{x-2}=\frac{1}{4}[/tex]We solve for x, like this:
[tex]\begin{gathered} \frac{x-2+x}{x\cdot(x-2)}=\frac{1}{4} \\ (2x-2)\cdot4=x\cdot(x-2) \\ 8x-8=x^2-2x \\ x^2-10x+8=0 \\ a=1 \\ b=-10 \\ c=8 \\ \text{ replacing in the general formula} \\ x_{1,2}=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x_1=\frac{-(-10)+\sqrt[]{(-10)^2-4\cdot1\cdot8}}{2\cdot1}=9.12 \\ x_2=\frac{-(-10)-\sqrt[]{(-10)^2-4\cdot1\cdot8}}{2\cdot1}=0.87 \end{gathered}[/tex]The value cannot be 0.87 because it is a value below 4 hours, therefore, the value that it has here is 9.17 hours
Therefore:
Jack = 9.12 hours
Lucy = 7.12 (9.12 - 2) hours
