How do you factor and solve $ {x^2} + 3x + 1 = 0 $ ?

Hint: Here, we are given the quadratic equation $ {x^2} + 3x + 1 = 0 $ . We can see that it cannot be solved by using simple factorization. For such quadratic equations that cannot be solved by factoring, we use a method which can solve all the quadratic equations known as completing the square method. Thus here, first we will convert the equation into a complete square and then solve it.

Complete step-by-step solution:
We are given $ {x^2} + 3x + 1 = 0 $ .
Our first step is to divide all the terms by the coefficient of $ {x^2} $ . But, here as the coefficient of $ {x^2} $ is 1, we can skip this step.
In the second step, we will move the constant term to the right side of the equation.
 $ \Rightarrow {x^2} + 3x = - 1 $
Now in the next step, we will complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation.
 $
   \Rightarrow {x^2} + 3x + {\left( {\dfrac{3}{2}} \right)^2} = - 1 - {\left( {\dfrac{3}{2}} \right)^2} \\
   \Rightarrow {x^2} + 3x + \dfrac{9}{4} = - 1 + \dfrac{9}{4} \\
   \Rightarrow {\left( {x + \dfrac{3}{2}} \right)^2} = \dfrac{5}{4} \\
  $
We will now take the square root of both the sides.
 $ \Rightarrow x + \dfrac{3}{2} = \pm \dfrac{{\sqrt 5 }}{2} $
Thus it can be written in the form of factors as:
 $ \Rightarrow \left( {x + \dfrac{3}{2} + \dfrac{{\sqrt 5 }}{2}} \right)\left( {x + \dfrac{3}{2} - \dfrac{{\sqrt 5 }}{2}} \right) = 0 $
Also, subtracting $ \dfrac{3}{2} $ from both the sides of equation $ x + \dfrac{3}{2} = \pm \dfrac{{\sqrt 5 }}{2} $ , we get
\[ \Rightarrow x = \dfrac{{\sqrt 5 }}{2} - \dfrac{3}{2}\] or \[x = - \dfrac{{\sqrt 5 }}{2} - \dfrac{3}{2}\]

Note: Here, we have solved the given quadratic equation by using completing the square method. In general, this method has five steps by which we can solve any quadratic equation.
Step 1. Divide all terms by the coefficient of $ {x^2} $
Step 2. Move the constant term to the right side of the equation.
Step 3. Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
Step 4. Take the square root on both sides of the equation.
Step 5. Subtract the number that remains on the left side of the equation to find the final value of $ x $