If one number is $80\%$ of the other and $4$ times the sum of their squares is $656$, then What are the numbers?
My answer is -
let the two numbers be x and y.
$x = \frac 45 y$
$4(x^2 + y^2) = 656$
$\implies x^2 + y^2 = 164$
Now substitute $x =\frac 45 y$:
\begin{align} &\implies \frac{16}{25}y^2 + y^2 = 164\\ &\implies \frac{41}{25}y^2 = 164 \\ &\implies y^2 = 100 \end{align} therefore $y$ can be $10$ or $-10$.
and $x$ can be the $8$ or $-8$.
However, the answer given is $8$, and $10$. My doubt is the answer should be $8,10$ and $-8$,$-10$. Am I correct here?
Yes, if there are no other conditions, {8,10} and {-8,-10} are both valid answers. As for percentages being negative, a percent symbol just means divide by 100. So -125.658% is valid and so is 3^(1/2)%