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= ${\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$

Now, x²-x+1 = 0

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

And x²+x+1 = 0

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

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

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

ALSO READ:

Solve x^3-1=0

Solve x^4+x^2+1=0

Solve x^4+4x^2+3=0

FAQs