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+2⋅x²⋅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.

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