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The difference of the squares of two positive numbers is $180$. The square of the smaller number is $8$ times the greater number. Find the two numbers.
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Solution: (a) Let the smaller number be $y$ and greater number be $x$.
A.T.Q.
$x^2 - y^2 = 180$
$y^2 = 8x$
$\Rightarrow x^2 - 8x = 180$
$x^2 - 8x - 180 = 0$
$(x - 18) (x + 10) = 0$
$x = 18, x = -10$ (rejected)
$\therefore$ The numbers are $18$ and $12$
A.T.Q.
$x^2 - y^2 = 180$
$y^2 = 8x$
$\Rightarrow x^2 - 8x = 180$
$x^2 - 8x - 180 = 0$
$(x - 18) (x + 10) = 0$
$x = 18, x = -10$ (rejected)
$\therefore$ The numbers are $18$ and $12$