Question

If “from the square of a half the sum of two numbers we subtract the square of a half their difference ”, the result is the :

(a) sum of the two numbers

(b) quotient of the two numbers

(c) difference of the two numbers

(d) product of the two numbers

Answer 1

Correct option is (d) product of the two numbers

Let two number x and y.

Now sum of these two numbers = x + y

Now Half of these two number = \(\frac{x +y}2\)

Now square will be = \(\left(\frac{x +y}2\right)^2\)

Now if we subtract x and y and than half of these numbers will be \(\frac{x -y}2\)

Square will be \(\left(\frac{x -y}2\right)^2\)

Now subtracting

⇒ \(\left(\frac{x +y}2\right)^2 - \left(\frac{x -y}2\right)^2\)

⇒ \(\frac{x^2 +2xy + y^2}4 - \frac{x^2 -2xy + y^2}4\)

⇒ \(\frac{x^2 +2xy + y^2 - x^2 + 2xy - y^2}4\)

⇒ \(\frac{4xy}4 = xy\)

Hence it will be the product of the two numbers.