Question

answer attempt 1 out of 2 write two numbers that multiply to the value on top and add to the value on bottom.

Explanation:

Step1: Let the two - numbers be \(x\) and \(y\).

We need \(xy\) (product) and \(x + y\) (sum) relationships. Let's assume the numbers satisfy \(xy=a\) and \(x + y=b\). Here, we want two numbers \(x\) and \(y\) such that \(xy\) gives the product of the non - diagonal numbers and \(x + y\) gives the sum of the non - diagonal numbers. Let the two numbers be \(x\) and \(y\), we know that we need to find two numbers that satisfy the condition related to 12 and 35.
We consider the quadratic equation \(t^{2}-(x + y)t+xy = 0\), or \(t^{2}-bt + a=0\). In our case, we want two numbers \(x\) and \(y\) such that \(xy=12\times35 = 420\) and we need to find two factors of 420 that add up to a certain value.
We factorize 420: \(420=2\times2\times3\times5\times7\).
We try to find two factors of 420 that add up to a reasonable value.
Let's find two numbers that multiply to 420. We know that \(20\times21 = 420\) and \(20+21=41\).
We can also think in another way. If we assume the two numbers \(x\) and \(y\) satisfy the system of equations based on the cross - multiplication and addition requirements.
Let's list factor pairs of 420: \((1,420),(2,210),(3,140),(4,105),(5,84),(6,70),(7,60),(10,42),(12,35),(14,30),(15,28),(20,21)\)
We check the sum of each pair.
We find that the two numbers are 20 and 21.

Answer:

20, 21