Check whether the following is a quadratic equation:
$(x-2)^{2}+1=2x-3$

Find two consecutive odd positive integers,the sum of whose squares is $290$.

$A$ cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was $3$ more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ $90,$ find the number of articles produced and the cost of each article.

Find the roots of the following equation:
$\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}, x \neq -4, 7$

Check whether the following is a quadratic equation:
$(x-2)(x+1)=(x-1)(x+3)$

Check whether the following is a quadratic equation:
$x^{2}+3x+1=(x-2)^{2}$