Blog Post # 4 Completing the Square

Today we will be completing the square.

We complete the square in order to solve a quadratic equation. 

Recall that the formula for a quadratic expression is ax² + bx +c².

In this lesson, we will be completing the square when the leading coefficient, “a”, is equal to 1.

Recall that a perfect square trinomial can be factored as (a+b)² = a² +2ab +b². 

For example, (x+4)² = x² + 8x + 16.

In many cases however, the expression doesn’t have a perfect square trinomial.

In x² + 8x – 33 = 0, we complete the square in order to get a perfect square trinomial. 

We do this by dividing “b” by 2 and squaring the answer [(b/2)²]. 

Here that is (8/2 = 4)²; 4² = 16. 

Once we have completed this step, we should isolate the “c” value.

x² + 8x – 33 +(33) = 0 + (33). 

Now we have x² + 8x = 33. 

We want to complete the square by converting “x² + 8x” into a perfect square trinomial. 

Again, to do this we take half of b and square it, [(b/2)²] = (8/2 = 4)²; 4² = 16. 

Now we add 16 to both sides of the equation.

x² + 8x + 16 = 33 + 16. 

This then equals x² + 8x + 16 = 49. 

Now we can factor the perfect square trinomial on the left hand side. 

From factoring, we get (x + 4)² = 49. 

We now have a quadratic that we can solve by taking a square root.

√(x + 4)² = √49. 

So, x+4 = 7 or x+4 = -7.  

We then solve both equations for x.

(x + 4) -4 = (7) -4 => x=3 and (x + 4) -4 = (-7) -4 => x= -11.

You can and should check your answers by plugging them back into x² + 8x + 16 = 49.