Question

How do you solve \[2{{x}^{4}}-{{x}^{2}}+3=0\] ?

Answer 1

Hint: In this question, we have to find the value of x. The equation given in the question is not in the form of the quadratic equation. Therefore, we will convert it into this form. After that we will apply splitting the middle method, to get two values of x. First, we will split the middle term of the equation and write it in factor form. After the necessary calculations, we get two values of x, which is our required solution.

Complete step by step answer:
Now in the question we have \[2{{x}^{4}}-{{x}^{2}}+3=0\].
Here we will assume,
\[\begin{align}
  & {{x}^{4}}={{x}^{2}} \\
 & \Rightarrow {{x}^{2}}=y \\
\end{align}\]
Thus, we get the equation in \[a{{x}^{2}}+bx+c=0\] form. After rearranging the equation, we get
\[2{{x}^{2}}-x+3=0\] .
As we can see that the above equation is in the form of \[a{{x}^{2}}+bx+c=0\] therefore, we will further solve this quadratic equation using either factorization method or by completing the square method.
Here we will use the factorization method. It is an easy method to solve quadratic equations.
First, we have to split the middle term \[bx\] in two terms, such that the product of the two terms is equal to the constant \[c\] and the sum of the two terms is equal to the middle term \[bx\]. Then factor the first two terms and the last two terms.
Here we can see the middle term is \[-x\] and the constant is 3 thus the equation can be written as
\[2{{y}^{2}}+2y-3y-3=0\]. We can see that the product of the two terms is equal to the constant \[-6\] and the sum of the two terms is equal to the middle term \[-x\]. Therefore, we will further factorize and get the roots.
\[\begin{align}
  & 2{{y}^{2}}+2y-3y-3=0 \\
 & \Rightarrow 2y(y+1)-3(y+1)=0 \\
\end{align}\]
Take \[(y+1)\] as a common factor.
\[\Rightarrow (y+1)(2y-3)=0\]
\[\Rightarrow y+1=0\] or \[2y-3=0\]
\[\Rightarrow y=-1,\dfrac{3}{2}\]
Thus \[-1,\dfrac{3}{2}\] are the extraneous roots of the given equation.
Now as according to our assumption \[{{x}^{2}}=y\]
Therefore, the solution is \[x=\sqrt{-1}.\sqrt{\dfrac{3}{2}}\]

Note:
 While finding the roots of the quadratic equation you can use completing the square method. After finding the roots of the assumed equation don’t forget to reverse your assumption to find the actual roots. Substitute the values in the original equation which is given in the question so, that you can verify your answer.