Question

the product of two consecutive integers is 90 find the two integers then check your answer?

Answer 1

Let n be the integer. Then, we can write the following equation

$n(n+1)=90$

which is equal to

$n^2+n-90=0$

We can solve this quadratic function by applying the quadratic formula:

$n=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}$

where in our case a=1, b=1 and c= -90. By substituting these values into the last formula, we get

$n=\frac{-1\pm\sqrt[]{1^2-4(1)(-90)}}{2(1)}$

then, we have

$\begin{gathered} n=\frac{-1\pm\sqrt[]{1+360}}{2} \\ n=\frac{-1\pm\sqrt[]{361}}{2} \\ n=\frac{-1\pm19}{2} \end{gathered}$

Then, the first solution is

$\begin{gathered} n=\frac{-1+19}{2} \\ n=\frac{18}{2} \\ n=9 \end{gathered}$

and the second solution is

$\begin{gathered} n=\frac{-1-19}{2} \\ n=-\frac{20}{2} \\ n=-10 \end{gathered}$

Then, we have the following solutions:

- Negative integer: n= - 10

- Positive integer: n=9