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Completing the Square

Q: What is completing the square?

A: Completing the square is a technique used to convert from general form, y=ax^2 + bx + c, to standard form, y=a(x-p)^2+q. Why do we need to convert? Standard form is much easier to graph as a parabola.

Let’s look at some examples:

Write each function in the form y=a(x-p)^2+q

a) y=x^2 + 10x + 30

Step 1: Take half of the b term (in this case, b=10) and square it.


Step 2: Add that number to the equation, but then subtract it from the end of the equation. Why? So that you are actually just adding 0 to the equation.

y=x^2 + 10x + 25 + 30 - 25

Step 3: You now have a perfect square (because 5+5=10, and 5*5=25), so group those terms together and factor:

y=(x^2 + 10x + 25) + 30 - 25
y=(x + 5)^2 + 30 - 25
y=(x + 5)^2 + 5Parabola y=(x+5)^2 +5

This is a parabola in standard form
with a=1, and the vertex is at (p,q)=(-5,5).

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