Completing the Square

There comes a time when we cannot factor a simple trinomial or complex trinomial because there is no combination of numbers that will multiply to get a product of the last term and add to get the sum of the middle term. If we are unable to do this we cannot find the x-ints resulting into not being able to find the vertex if required. This is when completing the square is used, so that we can go from standard form to vertex form, therefore being able to find the vertex.

When completing the square there are a few steps that we need to follow:

Let’s do an example…

Convert y = x^2 +4x + 6 to vertex form

Step #1: Group all the x^2 and x terms together.

y = [x^2 +4x] + 6

Step #2: Divide the middle term by 2 and then square that number. Make sure you add a negative off that number so you don’t change the equation.

y = [x^2 +4x] + 6y = [x^2 +4x+ 4 - 4] + 6

Step #3: As you can see in the above equation there is a perfect square. ( Recall (a+b)^2 =a^2 +2ab+b^2), so you can factor the first three terms as a perfect trinomial, while taking the negative number out of the brackets. When we do that we get…

y = [x^2 +4x+ 4] - 4 + 6

y = [x+ 4]^2 - 4 + 6

Step #4: Then we would just simplify the equation, and then we would have an equation in vertex form.

y = [x^2 +4x+ 4] – 2

Let’s try another example this time with an a value greater than 1 and a negative middle term……

Convert y = 3x^2 +6x + 9 to vertex form

Step #1: Common factor out the a value from all the terms.

y = 3x^2 -6x + 9

y = 3[x^2 -2x + 3]

Step #2: Group all the x^2 and x terms together.

y = 3[(x^2 - 2x) + 3]

Step #3: Divide the middle term by 2 and then square that number. Make sure you add a negative off that number so you don’t change the equation.

y = 3[(x^2 - 2x + 1 - 1) + 3]

Step #4: Factor the first three terms as a perfect trinomial, while taking the negative number out of the brackets. Make sure you multiply it with the number in front of the bracket when you take it out of the bracket.

y = 3(x- 1)^2 -3 + 3

y = 3(x- 1)^2

Now you try…

Convert the equation -24x = -3x^2 + 5 to vertex form (Hint: carry the -24x over to the other side.)

**(Hint: carry the -24x over to the other side.)**

Answer…

y = -3(x- 4)^2 + 53

Extra Practice…

A Video for Extra Help…