Quadratic Equations: Word Problems

The product of two consecutive natural numbers which are multiples of 3 is equal to 810. Find the two numbers.


Let the two consecutive natural numbers, which are multiples of 3, be $x$ and $x+3$

Then, as per question

$x \left(x+3\right) = 810$

i.e. $x^2 + 3x - 810 = 0$

i.e. $x^2 -27x + 30x -810 = 0$

i.e. $x \left(x-27\right)+30\left(x-27\right)=0$

i.e. $\left(x-27\right)\left(x+30\right)=0$

$\implies$ $x=27$ or $x=-30$

Since the required numbers are natural numbers, therefore, $x=27$ is the acceptable solution.

$\therefore$ The required numbers are 27, 30