Question

What two numbers have the product of 234 and the sum of 31

Answer 1

The answer is 13 and 18

Two numbers have the product of 234: x * y = 234
Two numbers have the sum of 31: x + y = 31

This is the system of equations:
x * y = 234
x + y = 31
_______
x * y = 234
y = 31 - x
_______
x(31 - x) = 234
31x - x² = 234      (-1)
-31x + x² = -234
x² - 31x = -234
x² - 31x + 234 = 0

According to the quadratic equation formula:
$x_{1,2} = \frac{-b+/- \sqrt{ b^{2}-4ac } }{2a} = \frac{-(31)+/- \sqrt{ (-31)^{2}-4*1*234 } }{2*1} = \\ \\ = \frac{31+/- \sqrt{ 961-936 } }{2} =\frac{31+/- \sqrt{25} }{2} = \frac{31+/-5}{2} \\ \\ x_1 =\frac{31+5}{2}= \frac{36}{2}=18 \\ \\ x_1 =\frac{31-5}{2}= \frac{26}{2}=13$

When: x1 = 18, y1 = 31 - 18 = 13
When: x2 = 13, y1 = 31 - 13 = 18