Solve the quadratic equation x^2 + 4x +1 = 0 by completing the square.
Completing the square means to put our equation into a slightly different form which looks like this, where a and b are real numbers:
(x+a)2 + b = 0
From here, we can rearrange the equation and directly solve for x. Let's have a look at our specific example:
x2 +4x +1 = 0
The first step is to divide the coefficient of x by 2, and add this to x (this is our value of 'a' to go inside our bracket). We then square this value of a and subtract it outside the bracket.
In our example it will look like this:
(x+2)2 - 4 + 1 = 0
(x+2)2 - 3 = 0
We have our equation in completed square form.
[There is a quick way to check we've got this right by expanding out this equation quickly:
(x+2)(x+2) - 3 = 0
x2 + 4x + 4 - 3 = 0
x2 + 4x +1 = 0
We're back to our original equation, so we know we've got it right. Let's go and solve our equation in completed square form.]
We simply rearrange for x:
(x+2)2 - 3 = 0
Add 3 to both sides.
(x+2)2 = 3
Take the square root of both sides. This splits into two possible cases:
Case 1: Positive square root of 3
x+2 = + sqrt(3)
x = - 2 + sqrt(3)
Case 2: Negative square root of 3
x+2 = - sqrt(3)
x = - 2 - sqrt(3)
So our final answer is...
x = - 2 + sqrt(3)
x = - 2 - sqrt(3)
