If sum of two numbers is 42 and their product is 437 , then find their difference.

Let’s denote the two numbers as \(x\) and \(y\). We are given that:

1. The sum of the two numbers is 42:
\[x + y = 42\]

2. Their product is 437:
\[xy = 437\]

We are asked to find their difference, which is \(|x – y|\).

To find \(x\) and \(y\), we can use the given sum and product. These can be considered as the sum and product of the roots of a quadratic equation, which can be written as:

\[x^2 – (x + y)x + xy = 0\]

Substituting the given values:

\[x^2 – 42x + 437 = 0\]

Now, we solve this quadratic equation for \(x\). The solutions to the equation will give us the values of \(x\) and \(y\). Instead of directly solving the quadratic equation, let’s use another approach to find the difference, as solving it might get a bit complex without a calculator.

The difference squared can be found using the formula derived from the sum and product of roots:
\[(x – y)^2 = (x + y)^2 – 4xy\]

Given \(x + y = 42\) and \(xy = 437\), we substitute these values into the equation:
\[(x – y)^2 = 42^2 – 4 \cdot 437\]

Calculating the right side:
\[(x – y)^2 = 1764 – 1748\]

\[(x – y)^2 = 16\]

Taking the square root of both sides:
\[|x – y| = 4\]

Therefore, the difference between the two numbers is \(\boldsymbol{4}\).