Question

The product of two consecutive positive integers is 306. Find the integers.

Answer 1

Hint: Here we will consider or assume two positive integers then after we will write an equation having this product as 306 and we will find the unknown and after that we will add one to find the final answer.
Formula used:
The general form of a quadratic equation is $ a{x^2} + bx + c = 0 $ and the values of $ x $ is given by the formula $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $

Complete step-by-step answer:
The product of two consecutive positive integers is 306. We need to find the integers so basically consecutive integers are numbers that follow each other in order. They have a difference of 1 between every two numbers. In a set of consecutive numbers, the mean and the median are equal. Hence, we will take an integer ‘x’ as first integer so the second consecutive integer can be a next number of the given number or the previous number of the given number respectively.
Let the number be $ x $ and $ x + 1 $ . Here we are given that the product of this number is 306. Hence, we have $ x(x + 1) = 306 $ .
 Hence let’s find the number. So we have $ {x^2} + x = 306 $
Making the RHS of the equation Zero we get,
 $ {x^2} + x - 306 = 0 $
 $ \Rightarrow 1 \times {x^2} + 1 \times x + ( - 306) = 0 $
The above expression is identical to the general form of a quadratic equation,
 $ a{x^2} + bx + c = 0 $
And comparing both of them we have,
 $ a = 1,b = 1,c = - 306 $
Using the quadratic formula we have,
 $ x = \dfrac{{ - 1 \pm \sqrt {{1^2} - 4 \times 1 \times ( - 306)} }}{{2 \times 1}} $
 $ \Rightarrow x = \dfrac{{ - 1 \pm \sqrt {1 + 1224} }}{{2 \times 1}} = \dfrac{{ - 1 \pm \sqrt {1225} }}{{2 \times 1}} $
 $ \Rightarrow x = \dfrac{{ - 1 \pm 35}}{{2 \times 1}} $
 $ \Rightarrow x = \dfrac{{ - 1 + 35}}{{2 \times 1}},\dfrac{{ - 1 - 35}}{{2 \times 1}} $
 $ \Rightarrow x = \dfrac{{34}}{2},\dfrac{{ - 36}}{2} $
 $ \Rightarrow x = 17, - 18 $
In question we are asked about positive integers and as one solution is negative we will discard it. Hence we will take 17 as x and the next consecutive number is taken as x+1 that is 17+1=18
Hence the required numbers are 17 and 18 respectively.
So, the correct answer is “17 and 18”.

Note: We can even take x and x-1 as both consecutive numbers and make an equation and solve for x. In this case we will get 18 as x and 17 as x-1 so we have the same set of numbers 17 and 18 respectively. You can try it out yourself in the same procedure.