Given:
The eqution is given as, x^2 + 4x + 1 = 0.
The objective is to solve the equation by compleing the square.
Consider the middle of the equation.
[tex]2\cdot a\cdot b=4x[/tex]Here, the value of a is x. Then, the value of b can be calculated as,
[tex]\begin{gathered} 2(x)\cdot b=4x \\ b=\frac{4x}{2x} \\ b=2 \end{gathered}[/tex]To complete the equation add +b^2 and -b^2 to the equation.
[tex]\begin{gathered} x^2+4x+2^2-2^2+1=0 \\ x^2+4x+2^2-4+1=0 \\ x^2+4x+2^2-3=0 \\ (x+2)^2-3=0 \\ (x+2)^2=3 \end{gathered}[/tex]Take square root on both sides, to solve the value of x,
[tex]\begin{gathered} \sqrt[]{(x+2)^2}=\sqrt[]{3} \\ x+2=\pm\sqrt[]{3} \\ x=\pm\sqrt[]{3}-2 \\ x=+\sqrt[]{3}-2\text{ and -}\sqrt[]{3}-2 \end{gathered}[/tex]Hence, the value of x are +√3-2 and -√3-2.
