For a diamond problem, what two numbers have a sum equal to 31 and a product equal to 234?

\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=\frac{-(-31)+/-\sqrt{25}}{2*1} =\frac{31+/-5}{2} \\\\ x_1=\frac{31+5}{2}=\frac{36}{2}\to\boxed{18} \\\\ x_2=\frac{31-5}{2}=\frac{26}{2}\to\boxed{13} \\\\ (x-18)(x-13)=0 \\\\ 1)x-18=0 \ => \boxed{x=18} \\\\ 2)x-13=0 \ => \boxed{x=13}"> a + b = 31 => a = b − 31 ​ ab = 234 b ∗ ( b − 31 ) = 234 b 2 − 31 b − 234 = 0 a = 1 b = − 31 c = − 234 Δ = ( − 31 ) 2 − 4 ∗ 1 ∗ ( − 234 ) = 961 − 936 = 25 x 1 ​ ; x 2 ​ = 2 ∗ 1 − ( − 31 ) + / − 25 ​ ​ = 2 31 + / − 5 ​ x 1 ​ = 2 31 + 5 ​ = 2 36 ​ → 18 ​ x 2 ​ = 2 31 − 5 ​ = 2 26 ​ → 13 ​ ( x − 18 ) ( x − 13 ) = 0 1 ) x − 18 = 0 => x = 18 ​ 2 ) x − 13 = 0 => x = 13 ​

Answered by mariamikayla | 2024-06-10

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 31 y − y 2 = 234 − y 2 + 31 y − 234 = 0
using a graph tool-----> to resolve the second order equation
see the attached figure
the solution is
13 an d 18

Answered by calculista | 2024-06-11

The two numbers that have a sum of 31 and a product of 234 are 13 and 18. This can be found by using the equations for their sum and product, leading to a solvable quadratic equation. Solving this equation reveals both numbers easily.
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Answered by calculista | 2024-10-01