Completing the Square - Concept
When a radical is not a perfect square (1, 4, 9, 16, ), estimating square roots is a valuable tool. When asked to estimate the value of a radical between two consecutive integers, find the two perfect squares that are slightly less and slightly more than the radicand. Also, remember that negative numbers do not have a real number square root.
You guys are going to learn about lots
of different ways to solve
Graphing is great in finding the X intercepts
except for sometimes it's not
very precise, especially if you have an
answer that's not a whole number.
Also, quadratic equation is good but sometimes
people make mistakes with the
Completing the square will always work.
Factoring doesn't always work.
That's why completing the square is a
good tool to have in your options for
how to solve quadratic equations.
Before we start talking about how to do
it, what I want to do is review what
you guys already know about perfect
square trinomials. They look like this.
Perfect square trinomials look like
A plus B equals A squared 2 A plus B squared
or A minus B squared equals A squared
minus 2AB plus B squared.
This is our goal.
Completing the square means taking some
trinomial and writing it like
this as a squared binomial.
Like, for example, if we have X squared
plus BX, we can complete the square
or turn it into what would be a factored
perfect square binomial by adding
B over 2. Squared.
It's going to make a lot more sense once
you guys start trying these actual
But if I can take half of this B term
and then square it and add it to this
original statement, I would have a
perfect square trinomial.
And we'll do some package problems
to have it make more sense.
Here's the step-by-step process
for how to complete the square.
First thing you want to do is rewrite the
trinomial in the form X squared
plus BX equals C. It's super important
that you notice that my leading
coefficient here is 1. If my number
there was like 4 or 10 or negative
6 or something I'd have to divide
all three terms by that number.
It's absolutely critical that your
leading coefficient is 1.
So you get your X squared term and your
X to the first power alone on one side
of the equals and your constant
goes on the other side.
The next thing you're going to do is take
half of your B term, square it, and
add the results to both
sides of the equation.
Like what I would do here is take half
of B. B over 2. Square that
quantity and then add it to both
sides of this equation.
After it's been squared.
So what I'm going to have is X plus half
of B squared is equal to C plus half
of B squared.
That will make a lot more sense once
we start working with real numbers.
I just wanted to show you what
this looks like with symbols.
The next thing I would do to finish completing
the square or to solve for X
would be to take the square
root of both sides.
So I would square root here and square root
here and then go through and solve
It looks really tricky here
when we're just looking at the
formulas and symbols, but once you guys
start trying some problems that
have real numbers it will start making
a lot more sense for you.