Question

The sum of two numbers is 17 and their product is 72. Find the smaller number.

Answer 1

Answer: 8

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Step-by-step 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

`m = (-b\pm√(b^2-4ac))/(2a)\\\\m = (-(-17)\pm√((-17)^2-4(1)(72)))/(2(1))\\\\m = (17\pm√(1))/(2)\\\\m = (17\pm1)/(2)\\\\m = (17+1)/(2) \ \text{ or } \ m = (17-1)/(2)\\\\m = (18)/(2) \ \text{ or } \ m = (16)/(2)\\\\m = 9 \ \text{ or } \ m = 8\\\\`

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.

Answer 2

The smaller number is 8

8+9=17
8x9=72