We will proceed to solve each case to determine the solution of the problem.
<u>case a)</u> ![x^{2}+2x+4=0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B2x%2B4%3D0)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![x^{2}+2x=-4](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B2x%3D-4)
Complete the square. Remember to balance the equation by adding the same constants to each side.
![x^{2}+2x+1=-4+1](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B2x%2B1%3D-4%2B1)
![x^{2}+2x+1=-3](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B2x%2B1%3D-3)
Rewrite as perfect squares
![(x+1)^{2}=-3](https://tex.z-dn.net/?f=%28x%2B1%29%5E%7B2%7D%3D-3)
![(x+1)=(+/-)\sqrt{-3}\\(x+1)=(+/-)\sqrt{3}i\\x=-1(+/-)\sqrt{3}i](https://tex.z-dn.net/?f=%28x%2B1%29%3D%28%2B%2F-%29%5Csqrt%7B-3%7D%5C%5C%28x%2B1%29%3D%28%2B%2F-%29%5Csqrt%7B3%7Di%5C%5Cx%3D-1%28%2B%2F-%29%5Csqrt%7B3%7Di)
therefore
case a) is not the solution of the problem
<u>case b)</u> ![x^{2}-2x+4=0](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x%2B4%3D0)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![x^{2}-2x=-4](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x%3D-4)
Complete the square. Remember to balance the equation by adding the same constants to each side.
![x^{2}-2x+1=-4+1](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x%2B1%3D-4%2B1)
![x^{2}-2x+1=-3](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x%2B1%3D-3)
Rewrite as perfect squares
![(x-1)^{2}=-3](https://tex.z-dn.net/?f=%28x-1%29%5E%7B2%7D%3D-3)
![(x-1)=(+/-)\sqrt{-3}\\(x-1)=(+/-)\sqrt{3}i\\x=1(+/-)\sqrt{3}i](https://tex.z-dn.net/?f=%28x-1%29%3D%28%2B%2F-%29%5Csqrt%7B-3%7D%5C%5C%28x-1%29%3D%28%2B%2F-%29%5Csqrt%7B3%7Di%5C%5Cx%3D1%28%2B%2F-%29%5Csqrt%7B3%7Di)
therefore
case b) is not the solution of the problem
<u>case c)</u> ![x^{2}+2x-4=0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B2x-4%3D0)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![x^{2}+2x=4](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B2x%3D4)
Complete the square. Remember to balance the equation by adding the same constants to each side.
![x^{2}+2x+1=4+1](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B2x%2B1%3D4%2B1)
![x^{2}+2x+1=5](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B2x%2B1%3D5)
Rewrite as perfect squares
![(x+1)^{2}=5](https://tex.z-dn.net/?f=%28x%2B1%29%5E%7B2%7D%3D5)
![(x+1)=(+/-)\sqrt{5}\\x=-1(+/-)\sqrt{5}](https://tex.z-dn.net/?f=%28x%2B1%29%3D%28%2B%2F-%29%5Csqrt%7B5%7D%5C%5Cx%3D-1%28%2B%2F-%29%5Csqrt%7B5%7D)
therefore
case c) is not the solution of the problem
<u>case d)</u> ![x^{2}-2x-4=0](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x-4%3D0)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![x^{2}-2x=4](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x%3D4)
Complete the square. Remember to balance the equation by adding the same constants to each side.
![x^{2}-2x+1=4+1](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x%2B1%3D4%2B1)
![x^{2}-2x+1=5](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x%2B1%3D5)
Rewrite as perfect squares
![(x-1)^{2}=5](https://tex.z-dn.net/?f=%28x-1%29%5E%7B2%7D%3D5)
![(x-1)=(+/-)\sqrt{5}\\x=1(+/-)\sqrt{5}](https://tex.z-dn.net/?f=%28x-1%29%3D%28%2B%2F-%29%5Csqrt%7B5%7D%5C%5Cx%3D1%28%2B%2F-%29%5Csqrt%7B5%7D)
therefore
case d) is the solution of the problem
therefore
<u>the answer is</u>
![x^{2}-2x-4=0](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x-4%3D0)