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# How do you find two consecutive, negative integers whose product is $156$?

Last updated date: 14th Jul 2024
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Hint: This question is a word problem based on application of quadratic equations. In this question we need to find two consecutive negative integers whose product is equal to $156$. To solve this question we need to know any one method for finding the root of a quadratic equation such as the quadratic formula. By consecutive in mathematics we mean two numbers whose difference is equal to $1$.

Complete step by step solution:
Let us try to solve this question in which we are asked to find two consecutive negative numbers such that their product is equal to $156$. To solve this question we first assume the negative integers equals their product to $156$. From doing this we get a quadratic equation and solve this by using the quadratic formula for finding the roots of the quadratic equation.
Let’s try to solve this question.
Suppose that the first negative integer to be $- n$.
Since both the negative integers are consecutive, so the other integer will be $- n + 1$.
Since we are given that the product of both the negative integers equal to $156$. So we get the given equation.
$( - n)( - n + 1) = 156$
${n^2} - n = 156$
${n^2} - n - 156 = 0$ $(1)$
Equation $(1)$is a quadratic equation. To solve this quadratic equation we will use the quadratic formula, which is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ for any quadratic equation $a{x^2} + bx + c = 0$. In our quadratic equation ${n^2} - n - 156 = 0$ , we have
$a = 1 \\ b = - 1 \\ c = - 156 \\$
Now putting this value in the formula we get a solution of our quadratic equation.
$n = \dfrac{{1 \pm \sqrt {{{( - 1)}^2} - 4 \cdot 1 \cdot ( - 156)} }}{{2 \cdot 1}} \\ n = \dfrac{{1 \pm \sqrt {1 + 624} }}{2} \\ n = \dfrac{{1 \pm \sqrt {625} }}{2} \\$
As we know that $\sqrt {625} = \pm 25$.
So we get
$n = \dfrac{{1 \pm 25}}{2}$
So $n = \dfrac{{1 + 25}}{2} = \dfrac{{26}}{2} = 13$ and $n = \dfrac{{1 - 25}}{2} = \dfrac{{ - 24}}{2} = - 12$.
We get the value of $n = 13$ and $n = - 12$ it means value of $- n = - 13$ and $- n = 12$. Since both the numbers are negative so $- n = 12$ is not possible. Hence the two consecutive negative integers are $- 12$ and $- 13$. Product of $- 12$ and $- 13$ is equal to $156$.

Note: To solve these types of questions we have to make an equation form the statement and act accordingly as the type of equation as in the above question it is quadratic. We can solve the quadratic equation by using the completing square method and factor method.