Question

# The product of two consecutive positive integers is 306. Form the quadratic equation to find x, if x denotes the smaller integer.

Hint: In this particular question assume any variable be the smallest positive integer so (variable + 1) is the consecutive positive integer greater than the variable so use these concepts to reach the solution of the question.

Let x be the smallest positive integer.

So, (x + 1) is the consecutive positive integer greater than x.

Now it is given that the product of two consecutive positive integers is 306.

So, first form the quadratic equation.

$\Rightarrow x\left( {x + 1} \right) = 306$

Now simplify the above equation we have,

$\Rightarrow {x^2} + x - 306 = 0$

So this is the required quadratic equation.

Now factorize this equation we have,

$\Rightarrow {x^2} + 18x - 17x - 306 = 0$

Now simplify the above equation we have,

$\Rightarrow x\left( {x + 18} \right) - 17\left( {x + 18} \right) = 0$

$\Rightarrow \left( {x + 18} \right)\left( {x - 17} \right) = 0$

$\Rightarrow \left( {x + 18} \right) = 0,{\text{ }}\left( {x - 17} \right) = 0$

$\Rightarrow x = 17, - 18$

So the positive integer is 17.

Therefore two consecutive positive integers are 17 and (17 + 1 = 18) whose product is 306.

Therefore the smallest positive integer is 17.

So this is the required answer.

Note: Whenever we face such types of questions the key concept is the construction of a quadratic equation according to given information in the problem statement then solve this quadratic equation as above and calculate the value of x which is the required smallest positive integer so that the product of two consecutive positive integer is 306.