Question

In an exam Ramya got $9$ marks less than twice that of Udaya got. Product of their marks is $195$. Who got more marks and how much?

Hint: Assume the marks of Udaya as a variable. Then by the information given you form equations and solve them to get marks of both.

Let marks of Udaya be $x$
This means marks of Ramya is $2x - 9$ where $2x$ is twice of Udaya’s marks and $9$ is subtracted as she got $9$ less marks than her twice
Product of their marks is $= 195$
Marks of Ramya$\times$Marks of Udaya $= 195$
$\Rightarrow (2x - 9) \times x = 195 \\ \Rightarrow 2{x^2} - 9x = 195 \\ \Rightarrow 2{x^2} - 9x - 195 = 0 \\$
This is a quadratic equation of the form $a{x^2} + bx + c = 0$
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
In this equation $a = 2\,,\,b = - 9\,,\,c = - 195$. So,
$x = \dfrac{{ - ( - 9) \pm \sqrt {{{( - 9)}^2} - 4 \times (2) \times ( - 195)} }}{{2 \times 2}} \\ x = \dfrac{{9 \pm \sqrt {81 + 1560} }}{4} \\ x = \dfrac{{9 \pm \sqrt {1641} }}{4} \\ x = \dfrac{{9 \pm 40.5}}{4} \\ x = \dfrac{{9 + 40.5}}{4},\dfrac{{9 - 40.5}}{4} \\ x = 12.375, - 7.875 \\$
$- 7.875$ won’t be considered as marks are not considered to be negative until given in the question.
Hence, is $x = 12.375$
$\Rightarrow 2x - 9 = 15.75$
Marks of Udaya are $12.375$
Marks of Ramya are $15.75$
Therefore, Ramya got more marks i.e. $15.75$

Note: As this was a question of marks, we ignored the negative solution obtained by solving the quadratic equation. But in other cases like that of temperature, we do need to consider the negative solution as well.