1. **State the problem:** Solve for $Y$ in the equation $(1.50Y+4)(0.50Y-4)=140$. 2. **Use the distributive property (FOIL) to expand:** $$ (1.50Y)(0.50Y) + (1.50Y)(-4) + 4(0.50Y) + 4(-4) = 140 $$ 3. **Calculate each term:** $$ 0.75Y^2 - 6Y + 2Y - 16 = 140 $$ 4. **Combine like terms:** $$ 0.75Y^2 - 4Y - 16 = 140 $$ 5. **Bring all terms to one side to set equation to zero:** $$ 0.75Y^2 - 4Y - 16 - 140 = 0 $$ $$ 0.75Y^2 - 4Y - 156 = 0 $$ 6. **Multiply entire equation by 4 to clear decimals:** $$ 3Y^2 - 16Y - 624 = 0 $$ 7. **Use quadratic formula:** $$ Y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=3$, $b=-16$, $c=-624$. 8. **Calculate discriminant:** $$ \Delta = (-16)^2 - 4(3)(-624) = 256 + 7488 = 7744 $$ 9. **Square root of discriminant:** $$ \sqrt{7744} = 88 $$ 10. **Calculate roots:** $$ Y = \frac{16 \pm 88}{6} $$ 11. **Find two solutions:** - $$ Y = \frac{16 + 88}{6} = \frac{104}{6} = \frac{52}{3} \approx 17.33 $$ - $$ Y = \frac{16 - 88}{6} = \frac{-72}{6} = -12 $$ **Final answer:** $$ Y = \frac{52}{3} \text{ or } Y = -12 $$