What do you do when you aren’t able to solve by factoring or using square roots?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
Thankfully, we can solve by completing the square!
When we are given a quadratic equation (polynomial of degree two), we can transform the equation through a series of steps so we are able to arrive at all possible roots.
So, what are the completing the square steps?
First, the leading coefficient must be a positive one. Therefore, it may be necessary to transform the given equation in order to create a positive one leading coefficient.
Next, we get to add in our boxes!
Yep, we’re completing the square so it’s only right that we have to plug in little squares into our equation!
Now we must determine the number that goes into these boxes. All we have to do is take half the coefficient of the middle term!
In doing so, we complete the box… we complete the square!
Cool, huh?!
Then, we factor!
But don’t worry, I’ve got a way to factor that feels like magic and will have us simplified in the blink of an eye.
Finally, we solve by using the square root. Never forgetting to add plus-or-minus to both sides.
Example for How to Complete the Square
Now at first glance, solving by completing the square may appear complicated, but in actuality, this method is super easy to follow and will make it feel just like a formula.
More importantly, completing the square is used extensively when studying conic sections, transforming integrals in calculus, and solving differential equations using Laplace transforms.
And did you know that without the method of completing the square we wouldn’t have the quadratic formula?
Yep, the quadratic formula is derived by completing the square!
Advanced Example for Complete the Square
It’s such a wonderful method and another tool in your mathematics toolbox, as Math is Fun so nicely states.
So, together we will work through countless examples for both expressions and also quadratic equations where we will steadily increase in level of difficulty.
Completing the Square (How-To) – Video
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