Find all real values of $a$ such that $x^4-2ax^2+x+a^2-a=0$

Question

I have to solve this question

Find all real values of $a$ such that $$x^4-2ax^2+x+a^2-a=0$$

I have been trying this question for many days but was unable to solve it.

I think we can convert this polynomial into a perfect square of some other polynomial or we can convert it into a quadratic polynomial and the do $D\geq 0$ so we can find a range bound in a and then can give the answer may be.

Or another method is we can try to factorise it by getting a value of x and then we can maybe further solve.

I have tried all methods but was unable to crack the problem. I think the quadratic method is best but I am unable to get any result from that.

So please help me by giving me approach to this question and also share the way why you used this particular method only and not the other ones and what approach one should develop while doing these questions.

Thanks

Answer 1

Your equation$$x^4-2ax^2+x+a^2-a=0$$is a quadratic equation in $a$:$$a^2-(1+2x^2)a+x^4+x=0$$and therefore you can apply the quadratic formula. Note that$$(1+2x^2)^2-4(x^2+x)=4x^2-4x+1=(2x-1)^2.$$Therefore, the solutions are $a=x^2-x+1$ and $a=x^2+x$.

Answer 2

You can factorise your expression into two second order polynomials $$x^4-2ax^2+x+a^2-a$$$$=(-x^2-x+a)(-x^2+x-1+a)=0$$ and you can use the quadratic equation as needed.