**Example 1:**

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 ₹750. We would like to find out the number of toys produced on that day.

**Solution:**

**(i)** Let us say the number of marbles John has = *x*

Therefore, the number of marbles Jivanti has = 45 – *x*

After losing 5 marbles each,

Number of marbles John has = *x* – 5

Number of marble Jivanti has = 45 – *x* – 5 = 40 –* x*

Given that the product of their marbles is 124.

∴ (*x *– 5)(40 – *x*) = 124

⇒ *x*^{2} – 45*x* + 324 = 0

⇒ *x*^{2} – 36*x* – 9*x* + 324 = 0

⇒ *x*(*x* – 36) -9(*x* – 36) = 0

⇒ (*x* – 36)(*x* – 9) = 0

Thus, we can say,

*x* – 36 = 0 or *x* – 9 = 0

⇒ *x* = 36 or *x* = 9

Therefore,

If John’s marbles = 36

Then, Jivanti’s marbles = 45 – 36 = 9

And if John’s marbles = 9

Then, Jivanti’s marbles = 45 – 9 = 36

**(ii)** Let us say the number of toys produced in a day is *x*.

Therefore, cost of production of each toy = Rs(55 – *x*)

Given the total cost of production of the toys = Rs 750

∴ *x*(55 – *x*) = 750

⇒ *x*^{2} – 55*x* + 750 = 0

⇒ *x*^{2} – 25*x* – 30*x* + 750 = 0

⇒* x*(*x* – 25) -30(*x* – 25) = 0

⇒ (*x* – 25)(*x* – 30) = 0

Thus, either *x* -25 = 0 or *x* – 30 = 0

⇒ *x* = 25 or *x* = 30

Hence, the number of toys produced in a day will be either 25 or 30.