Find two consecutive positive integers, sum of whose squares is $365$.

Find the roots of the following quadratic equations by factorisation: (iii) $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$

Find the roots of the following quadratic equations by factorisation: (ii) $2x^2 + x – 6 = 0$

Find two numbers whose sum is $27$ and product is $182$.

Find the roots of the following quadratic equations by factorisation: (i) $x^2 – 3x – 10 = 0$

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 quadratic equations by factorisation: (iv) $2x^2 – x + \frac{1}{8} = 0$

Solve the problems given below.

Represent the following situations mathematically:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and
the product of the number of marbles they now have is 124. We would like to find
out how many marbles they had to start with.
(ii) A cottage industry produces a certain number of toys in a day. The cost of
production of each toy (in rupees) was found to be 55 minus the number of toys
produced in a day. On a particular day, the total cost of production was
Rs.750. We would like to find out the number of toys produced on that day.

The altitude of a right triangle is $7$ cm less than its base. If the hypotenuse is $13$ cm, find the other two sides.

Find the roots of the following quadratic equations by factorisation: (v) $100x^2 – 20x + 1 = 0$