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Completing the Square - Concept
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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

quadratic equations.

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

negative signs.

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.

That's tricky.

It's going to make a lot more sense once

you guys start trying these actual

practice problems.

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

for X.

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.

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