Algebra and geometry are closely connected. Geometry, as in coordinate graphing and polygons, can help you make sense of algebra, as in quadratic equations. Completing the square is one additional mathematical tool you can use for many challenges:
When completing the square, we can take a quadratic equation like this, and turn it into this:
"Completing the square" comes from the exponent for one of the values, as in this simple binomial expression:
We use for the second term because we reserve for the first one. We might have had , but if is 1, you have no need to write it.
Anyway, you have no idea what values or have, so how can you proceed? You already know will be multiplied times itself, to begin.
Think about a square in geometry. You have four congruent-length sides, with an enclosed area that comes from multiplying a number times itself. In this expression, times is a square with an area of :
Hold on -- we still have unknown variable times . What would that look like? That would be a rectangle units tall and units wide, attached to our square:
To make better sense of that rectangle, divide it equally between the width and length of the square. That would make each rectangle times :
That means the new almost-square is , but we are missing a tiny corner, which would have a value of times itself, or :
That last step literally completed the square, so now we have this:
This refines or simplifies to:
You need to also subtract if you are, in fact, trying to work an equation (you cannot add something without balancing it by subtracting it). In our case, we were just showing how the square is really a square, in a geometric sense.
Here is a more complete version of the same thing:
As soon as you see raised to a power, you know you are dealing with a candidate for "completing the square."
The role of from our earlier example is played here by the . We added a value, , so now we have a trinomial expression.
is rewritten as:
So, divide by and square it, which you then add and subtract to get:
Now, you can simplify as:
Which is equal to:
This simplifies to:
On a graph, this plots a parabola with a vertex at .
You can use completing the square to simplify algebraic expressions. Here is a straightforward example with steps:
Divide the middle term, , by and square it, then both add and subtract it:
Simplify the expression:
Seven steps are all you need to complete the square in any quadratic equation. The general form of a quadratic equation looks like this:
Completing The Square Steps
We will provide three examples of quadratic equations progressing from easier to harder. Give each a try, following the seven steps described above. The first one does not place a coefficient with :
Our second example uses a coefficient with for solving a quadratic equation by completing the square:
Our third example is all bells and whistles with really big numbers. See how you do!
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