A positive real number is 4 less than another. If the sum of the squares of the two numbers is 72, then find the numbers.

Our two numbers are:

Or, approximately 7.66 and 3.66.

Step-by-step explanation:

Let the two numbers be a and b.

One positive real number is four less than another. So, we can write that:

The sum of the squares of the two numbers is 72. Therefore:

Substitute:

Solve for a. Expand:

Simplify:

Divide both sides by two:

Subtract 36 from both sides:

The equation isn't factorable. So, we can use the quadratic formula:

In this case, a = 1, b = -4, and c = -28. Substitute:

Evaluate:

So, our two solutions are:

Since the two numbers are positive, we can ignore the second solution.

So, our first number is:

And since the second number is four less, our second number is:

Step-by-step explanation:

Let the large number be . We can represent the smaller number with . Since their squares add up to 72, we have the following equation:

Expand using the property :

Combine like terms:

Subtract 72 from both sides:

Use the quadratic formula to find solutions for :

for

In , assign:

Solving, we get:

Since the question stipulates that is positive, we have . Therefore, the two numbers are and .

Verify:

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step-by-step explanation: