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Ontario Tech acknowledges the lands and people of the Mississaugas of Scugog Island First Nation.

We are thankful to be welcome on these lands in friendship. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. These lands remain home to many Indigenous nations and peoples.

We acknowledge this land out of respect for the Indigenous nations who have cared for Turtle Island, also called North America, from before the arrival of settler peoples until this day. Most importantly, we acknowledge that the history of these lands has been tainted by poor treatment and a lack of friendship with the First Nations who call them home.

This history is something we are all affected by because we are all treaty people in Canada. We all have a shared history to reflect on, and each of us is affected by this history in different ways. Our past defines our present, but if we move forward as friends and allies, then it does not have to define our future.

Learn more about Indigenous Education and Cultural Services

Completing the Square

Completing the square is a process that helps us rewrite a quadratic that's in the form ax2 + bx + c into the form a(x -  h)2+k. So, why would we want to do this? Well, for starters, if you want to graph the parabola, a(x - h)2+k is a form that's easier to work with because you automatically know that the vertex is at (h, k).

(Also, if you're taking calculus, then knowing how to complete the square may help you integrate a complicated function; or, if you're taking a course in differential equations, it may help you to find the transformation of a function).

How to Complete the Square

Starting with ax2 + bx + c, here's how to complete the square:

Completing the square

OK, it's quite tricky to make sense of this just by looking at the steps, so hopefully the examples should make a lot more sense!

Example: Complete the square to write x2 + 6x + 11 in the form of a(x - h)2+k.

Solution:

x2 + 6x + 11

=(x2 + 6x + 9 - 9) + 11

=(x2 + 6x + 9) + 11 - 9

=(x2 + 6x + 9) + 2

= (x + 3)2 - 2

Example: Complete the square to write 2x2 - 16x + 10 in the form a(x - h)2+k.

2x2 - 16x + 10

= 2(x2 - 8x) + 10

= 2(x2 - 8x + 16 - 16) + 10

= 2(x2 - 8x + 16) -32 + 10

= 2(x - 4)2 -22

An Important Note:

Remember that if you expand the expression that you get for your final answer, you'll get the original expression in the question...this is a useful way to check your work!

Example 1:

Example 2:

Example 3: