Two numbers when multiplied give 72 and added give 17

Steps:

1. Let the two numbers be $x$ and $y$.
2. From the problem statement, we have two equations:
   • $x \cdot y = 72$ (Product)
   • $x + y = 17$ (Sum)
3. From the second equation, express $y$ in terms of $x$: $y = 17 - x$
4. Substitute this expression for $y$ into the first equation: $x \cdot (17 - x) = 72$ $17x - x^2 = 72$
5. Rearrange the terms to form a standard quadratic equation: $x^2 - 17x + 72 = 0$
6. Factor the quadratic equation. We need two numbers that multiply to 72 and add to -17. These numbers are -8 and -9. $(x - 8)(x - 9) = 0$
7. Solve for $x$: $x - 8 = 0 \Rightarrow x = 8$ $x - 9 = 0 \Rightarrow x = 9$
8. Find the corresponding $y$ values using $y = 17 - x$:
   • If $x = 8$, then $y = 17 - 8 = 9$.
   • If $x = 9$, then $y = 17 - 9 = 8$.

Both pairs (8, 9) and (9, 8) satisfy the conditions. Therefore, the two numbers are 8 and 9.