How to Factorise and Solve x^4+x^2+1=0

In today’s article, we will first factorize the expression x⁴+x²+1, and then we will solve the bi-quadratic equation x⁴+x²+1=0.

How to Factorise x⁴+x²+1

Answer: The factorisation of x⁴+x²+1 is given by x⁴+x²+1 = (x²-x+1)(x²+x+1).

Solution:

At first, we will add and subtract x² to the given expression.

x⁴+x²+1

= (x⁴+x²+1) + x² – x²

= (x⁴+2x²+1) – x²

= {(x²)² +2⋅x²⋅1+1²) – (x)²

Applying the formula a²+2ab+b² = (a+b)², we get

= (x²+1)² – (x)²

= (x²+1-x)(x²+1+x) as we know that a²-b² = (a-b)(a+b)

= (x²-x+1)(x²+x+1)

Video Solution on How to Factorise x⁴+x²+1:

Solve x⁴+x²+1=0

x⁴+x²+1= 0

⇒ (x²-x+1)(x²+x+1) = 0 by above.

Therefore, x²-x+1 = 0 or x²+x+1=0.

Apply the rule: ax²+bx+c = 0 ⇒ x= $\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

Now, x²-x+1 = 0

⇒ x = $\dfrac{1 \pm \sqrt{1-4}}{2}$ = $\dfrac{1 \pm \sqrt{3}i}{2}$

And x²+x+1 = 0

⇒ x = $\dfrac{-1 \pm \sqrt{1-4}}{2}$ = $\dfrac{-1 \pm \sqrt{3}i}{2}$

So the solutions of x⁴+x²+1= 0 are given by x= $\dfrac{1 + \sqrt{3}i}{2}$, $\dfrac{1 – \sqrt{3}i}{2}$, $\dfrac{-1 + \sqrt{3}i}{2}$ and $\dfrac{-1- \sqrt{3}i}{2}$.

So there are four solutions of the given equation x⁴+x²+1= 0.

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