# Factorise x^4+4x^2+3 | How to Solve x^4+4x^2+3=0

The equation x4+4x2+3=0 is a bi-quadratic equation. In this post, we will first factorise x4+4x2+3 and then solve the equation x4+4x2+3=0.

## How to Factorise x4+4x2+3

Question: Factorize x4+4x2+3.

Solution:

To factorize x4+4×2+3, our aim is to express it in the form of a2-b2. For that, we need to add and subtract 1 to both sides. So we have

x4+4x2+3

= x4+4x2+3 + 1 -1

= x4+4x2+4 – 1

= (x2)2+2x2⋅2+22 – 12

= (x2+2)2 – 12 as we know that a2+2ab+b2 = (a+b)2

= (x2+2-1) (x2+2+1) using the formula a2-b2 = (a-b)(a+b)

Simplifying the above, we obtain that

x4+4x2+3 = (x2+1) (x2+3) …(∗)

Thus, the factorization of x4+4x2+3 is given by x4+4x2+3 = (x2+1) (x2+3).

You can read: How to solve linear equations

## Solve x4+4x2+3=0

Question: Solve the equation x4+4x2+3=0.

Solution:

x4+4x2+3=0

⇒ (x2+1) (x2+3)=0, follows from above (∗)

We know that if the product of two numbers is zero, then the numbers are individually zero. Thus, we have

x2+1=0 or x2+3=0

⇒ x2-i2=0 or x2-3i2=0 where i=√-1 is an imaginary complex number, so i2=-1.

⇒ x2 = i2 or x2 = 3i2

Taking square root on both sides, we get that

x = i, -i or x = $\sqrt{3}$i, -$\sqrt{3}$i

So the solutions of x4+4x2+3=0 are given by x=±i, ±$\sqrt{3}$i.