The difference of squares of two numbers is 180. The square

(Q) The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Ans: Let the smaller number be x and larger number be y.

By 1st condition, $y^2 - x^2 = 180$ …. (i)

By 2nd condition, $x^2 = 8 y$ … (ii)

By substituting values of $x^2$ in equation (i), we get:

$\Rightarrow y^2 - 8 y = 180$

$\Rightarrow y^2 - 8 y - 180 = 0$

$\Rightarrow y^2 - 18 y + 10 y - 180 = 0$

$\Rightarrow y(y - 18) + 10(y - 18) = 0$

$\Rightarrow (y - 18) (y + 10) = 0$

$\Rightarrow y = 18 ~ and ~ y = - 10$

Here, if we take y = – 10, and substitute in equation (in), we get:

$x^2 = 8 y$

$\Rightarrow x^2 = 8 (- 10)$

$\Rightarrow x^2 = - 80$

Since, we can’t take square root of negative number, hence we reject y = -10 and we accept y = 18.

In equation (ii), we substitute y = 18, we get:

$x^2 = 8 y$

$\Rightarrow x^2 = 8 (18)$

$\Rightarrow x^2 = 144$

$\Rightarrow x = 12$

Hence, smaller number be 12 and larger number be 18.

Check:

If our numbers are 12 and 18, then by 1st condition,

$(18)^2 - (12)^2 = 324 - 144 = 180$

Since it meets the given condition, hence our answer is correct.

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