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Explanation:
Through guess and check, we find that
8+9 = 17
8*9 = 72
We see that 8 is the answer.
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Another approach:
Let m and n be the two numbers.
They sum to 17 which leads to m+n = 17. Solving for n gets us n = -m+17.
Their product is 72, so, m*n = 72
Apply substitution and we get the following:
m*n = 72
m*(-m+17) = 72
-m^2 + 17m = 72
-m^2 + 17m - 72 = 0
-1(m^2 - 17m + 72) = 0
m^2 - 17m + 72 = 0
Now apply the quadratic formula
[tex]m = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\m = \frac{-(-17)\pm\sqrt{(-17)^2-4(1)(72)}}{2(1)}\\\\m = \frac{17\pm\sqrt{1}}{2}\\\\m = \frac{17\pm1}{2}\\\\m = \frac{17+1}{2} \ \text{ or } \ m = \frac{17-1}{2}\\\\m = \frac{18}{2} \ \text{ or } \ m = \frac{16}{2}\\\\m = 9 \ \text{ or } \ m = 8\\\\[/tex]
If m = 8, then n = -m+17 = -8+17 = 9.
If m = 9, then n = 8 through similar steps.
We see the two values are 8 and 9, the smaller of which is 8.
