Solving quadratic equation using completing the square (Algebra 2)

It should not come as a shock to you that there are other methods of solving the quadratic equation.  In this article, we will discuss another method to solve the quadratic equation using the method called "completing the square".  Like I have said in my previous post, there is no one size fits all solution, which means that you can use different ways to solve a quadratic equation.

In this article, we will solve the quadratic equation of

x^2 + 6x - 7 = 0

Step #1: It is ideal to change original equation from

Ax^2 + Bx + C = 0

to

Ax^2 + Bx = C

So for our example equation, we will change it to

x^2 + 6x  = 7

Step 2:  Determine the new factor, by dividing the coefficient of B by 2 and squaring the result.  Add this new factor into both sides of the equation.

Since B = 6, sq(B/2) = sq(6/2) = sq(3) = 9, let's add sq(3) to both sides of the equation (remember in order to keep an equation equal, whatever is done on the left side of the equation, the same operation needs to happen on the right side).

x^2 + 6x + sq(3) = 7 + sq(3)

The beautiful thing about "completing the square" method is that we do not need to wreck our brain to figure out the factors on the left-hand side of the equation above. 

The left side of the equation is just

(x + 3)^2 = 7 + 9

In order to factor the left side of the equation, it is just taking one of the x term from Ax^2, and adding/subtracting half of the coefficient of B, and squaring the entire expression. 

Let's continue to solve this equation:

(x + 3)^2 = 7 + 9
(x + 3)^2 = 16
x + 3 = +/- 4
x = +/-4 - 3

So the 2 solutions to this quadratic equation is

x = +4 - 3 = 1

and

x = -4 - 3 = -7

Here are some more problems that you should use to practice on the completing the square method.

1. 4x^2 - 2x - 5 = 0
2. x^2 + 6x + 10 = 0 (note, this one provides imaginary roots, which means that the parabola does not have any x-intercepts)
3. x^2 - 2x - 48 = 0