Question

Solve the equation:
\[\left( {{y}^{2}}+5y \right)\left( {{y}^{2}}+5y-2 \right)-24=0\]

Answer 1

- Hint: -The procedure for solving this question is as follows
On closely observing the given equation, it can be written again with \[{{y}^{2}}+5y=a\] .
 Then, we can solve the problem accordingly.
After that, we can again enter the value of a then solve for y.

Complete step-by-step solution -

As mentioned in the question, we have to find all the roots of the polynomial of degree 4; we have to follow the procedure mentioned in the hint.
Now, we can write the given equation as follows using the procedure given in hint
\[\begin{align}
  & \left( {{y}^{2}}+5y \right)\left( {{y}^{2}}+5y-2 \right)-24=0 \\
 & \left( a \right)\left( a-2 \right)-24=0 \\
 & {{a}^{2}}-2a-24=0 \\
 & a=\dfrac{-(-2)\pm \sqrt{{{(-2)}^{2}}+4\times 24}}{2} \\
 & a=\dfrac{+2\pm 10}{2} \\
 & a=6,-4 \\
\end{align}\]
Now, as we know the value that we assigned to a, we get the following result
For a=6, we get the following
\[\begin{align}
  & {{y}^{2}}+5y=6 \\
 & {{y}^{2}}+5y-6=0 \\
 & y=\dfrac{-5\pm \sqrt{{{5}^{2}}+4\times 6}}{2} \\
 & y=\dfrac{-5\pm \sqrt{25+24}}{2} \\
 & y=1,-6 \\
\end{align}\]
And for a=-4, we get the following
\[\begin{align}
  & {{y}^{2}}+5y=-4 \\
 & {{y}^{2}}+5y+4=0 \\
 & y=\dfrac{-5\pm \sqrt{{{5}^{2}}-4\times 4}}{2} \\
 & y=\dfrac{-5\pm \sqrt{25-16}}{2} \\
 & y=-4,-1 \\
\end{align}\]
Hence, the roots of the polynomial with degree 4 are 1, -1, -6, and -4.

Note: -Another method of doing this question is as follows
This type of question can be solved by the following procedure:-
First, we will open up all the brackets and then we will get a polynomial of degree 4.
Now, we will try to find one or two roots of the obtained polynomial of degree 4.
After that, we will find the remaining roots by just dividing the polynomial of degree 4 with its factors that we will get on using the zeros of the polynomial of degree 4.