x^2+4x+3=0

Expert intro: This quadratic can be solved efficiently by completing the square, which rewrites the expression into a perfect-square form.

Detailed walkthrough

Step 1: Start with the equation

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

Move the constant term:

$x^2+4x=-3$

Step 2: Complete the square

Take half of 4, which is 2, and square it:

$2^2=4$

Add 4 to both sides:

$x^2+4x+4=-3+4$

Step 3: Rewrite as a square

$(x+2)^2=1$

Step 4: Solve

Take the square root of both sides:

$x+2=\pm 1$

So,

$x=-2+1=-1$

or

$x=-2-1=-3$

Answer

$\boxed{x=-1,-3}$

💡 Pitfall guide

A common mistake is adding 4 to one side but forgetting to add it to the other side. Another error is writing $(x+2)^2=7$ from the original equation; the correct result after completing the square is $(x+2)^2=1$.

🔄 Real-world variant

If the equation were $x^2+4x=3$, the same method would give $x^2+4x+4=7$, so $(x+2)^2=7$ and $x=-2\pm\sqrt7$. The setup changes only because the constant term is on the other side.

🔍 Related terms

completing the square, quadratic equation, perfect square trinomial