# The product of two consecutive positive integers is ${\text{306}}$, the quadratic equation to find the integers, if $x$ denotes the smaller integer is ${x^2} + x - 306 = 0$. Check whether the given statement is True or False?

$(a){\text{ }}$True

$(b){\text{ }}$False

Answer

Verified

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Hint: Be careful while assuming the numbers, here, it is given that the numbers are consecutive, the other cases are also possible.

Consider the given equation

\[{x^2} + x - 306 = 0\]

It is given that the smaller integer is \[x\],

Therefore, the other consecutive integer can be assumed to be \[{\text{ }}x + 1\].

Since the product of the two consecutive integers is \[{\text{306}}\].

Therefore, we have,

\[x(x + 1) = 306\]

\[{x^2} + x = 306\]

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

That is, the product of the two consecutive integers represent the quadratic equation

\[{x^2} + x - 306 = 0\]

So, the required answer is $(a){\text{ }}$True

Note: In these types of questions, the integers are first put into the respective conditions, and then solved for the required equation.

Consider the given equation

\[{x^2} + x - 306 = 0\]

It is given that the smaller integer is \[x\],

Therefore, the other consecutive integer can be assumed to be \[{\text{ }}x + 1\].

Since the product of the two consecutive integers is \[{\text{306}}\].

Therefore, we have,

\[x(x + 1) = 306\]

\[{x^2} + x = 306\]

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

That is, the product of the two consecutive integers represent the quadratic equation

\[{x^2} + x - 306 = 0\]

So, the required answer is $(a){\text{ }}$True

Note: In these types of questions, the integers are first put into the respective conditions, and then solved for the required equation.

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