Question

Question 7
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.

Answer 1

Let the larger and smaller number be x and y respectively.
According to the question,
$x^2-y^2=180and{y}^2=8x\Rightarrow x^2-8x=180\Rightarrow x^2-8x-180=0\Rightarrow x^2-18x+10x-180=0\Rightarrow x(x-18)+10(x-18)=0\Rightarrow (x-18)(x+10)=0\Rightarrow x=18,-10$
However, the larger number cannot be negative as 8 times of the larger number will be negative and hence, the square of the smaller number will be negative which is not possible.
Therefore, the larger number will be 18.
$x=18\therefore y^2=8x=8\times 18=144\Rightarrow y=\pm \sqrt{144}=\pm 12\therefore Smallernumber=\pm 12$
Therefore, the numbers are 18 and 12 or 18 and - 12.