If the difference of the squares of two numbers is 45, the square of the smaller number is 4 times the larger number, then the numbers are:

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9, 6 or 9, -6

5, 6 or 5,-6

9, 5 or 9, -5

None of these

answer is A.

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Detailed Solution

Concept-Start with the numbers x and y. Where

. Use the statement that the difference between the squares of the numbers is 45 to form the first equation, and use the statement that the square of the small number is four times the large number to form the second equation. Solve the equation to get the answer.
To determine the numbers, use the question statements to create equations and solve them. First, we consider the two numbers as x and y, so

. Next, let's create an equation using the information
in the question. The first statement shows that the difference between the squares of the two numbers is
45.
Therefore,

........ (i)
The second statement shows that the square of a small number is four times the number of a large number.
Therefore,

......... (ii)
Now, substitute the value of

in equation (ii) into equation (i). Substituting the value of

in equation (ii) into equation (i) yields:

Therefore,

....... (iii)
It is clear that equation (iii) is x and is a quadratic equation. Then use the mean term decomposition to solve the square. If you divide the intermediate term into two numbers so that the sum is

and the product

, you get:

Summarizing x from the first two terms and summarizing 5 from the last two terms gives:

Now the square of the second statement in the small number is four times the square in the large number.
Also, the square of any number is always positive, so large numbers cannot be negative. Therefore, small numbers cannot be negative. Therefore,

Therefore,

Now, if we insert

into equation (ii),

Hence, option 1 is correct

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