Answer: 2 and 9
2*9 = 18
2+9 = 11
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Explanation:
You can find this through trial-and-error. You're basically looking for factors of 18 which add to 11. Something like 6 and 3 won't work since 6+3 = 9, but 2 and 9 works since 2+9 = 11.
Or you can use algebra as shown below.
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Let x and y be the two numbers. They have a product of 18, so x*y = 18.
They also have a sum of 11, meaning x+y = 11. Solving for y leads to y = 11-x
Plug this into xy = 18 and we get x(11-x) = 18 which becomes -x^2+11x = 18
Now subtract 18 from both sides to arrive at the equation -x^2+11x-18 = 0.
Let's use the quadratic formula to solve.
We'll plug in a = -1, b = 11, c = -18
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(11)\pm\sqrt{(11)^2-4(-1)(-18)}}{2(-1)}\\\\x = \frac{-11\pm\sqrt{49}}{-2}\\\\x = \frac{-11\pm7}{-2}\\\\x = \frac{-11+7}{-2}\ \text{ or } \ x = \frac{-11-7}{-2}\\\\x = \frac{-4}{-2}\ \text{ or } \ x = \frac{-18}{-2}\\\\x = 2\ \text{ or } \ x = 9\\\\[/tex]
If x = 2, then y = 11-x = 11-2 = 9
If x = 9, then y = 11-x = 11-9 = 2
Both values of x lead to the same pair of values overall, meaning that 2 and 9 are the two values we're after.
2*9 = 18
2+9 = 11
