Question

For a diamond problem what two numbers' sums equal 31 and their product is 234?

Answer 1

`a+b=31=> \boxed{a=b-31} \\ab=234 \\\\ \\\\ b*(b-31)=234 \\\\ b^2-31b-234=0 \\ a=1 \\ b=-31 \\ c=-234 \\\\ \Delta= (-31)^2-4*1*(-234)= 961-936=25 \\\\ x_1;x_2=(-(-31)+/-√(25))/(2*1) =(31+/-5)/(2) \\\\ x_1=(31+5)/(2)=(36)/(2)\to\boxed{18} \\\\ x_2=(31-5)/(2)=(26)/(2)\to\boxed{13} \\\\ (x-18)(x-13)=0 \\\\ 1)x-18=0 \ => \boxed{x=18} \\\\ 2)x-13=0 \ => \boxed{x=13}`

Answer 2

Let

x-------> the first number

y------> the second number

we know that

`x+y=31`

`x=31-y`

equation
`1`

`x*y=234`

equation
`2`

substitute equation 1 in equation 2

`(31-y)*y=234\\ 31y-y^(2) =234\\ -y^(2) +31y-234=0`

using a graph tool-----> to resolve the second order equation

see the attached figure

the solution is

`13 and 18`