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### Course: Algebra 1 (Eureka Math/EngageNY) > Unit 4

Lesson 9: Topic B: Lessons 11-13: Completing the square- Completing the square
- Solving quadratics by completing the square
- Worked example: Completing the square (intro)
- Completing the square (intro)
- Worked example: Rewriting expressions by completing the square
- Worked example: Rewriting & solving equations by completing the square
- Completing the square (intermediate)
- Worked example: completing the square (leading coefficient ≠ 1)
- Completing the square
- Solving quadratics by completing the square: no solution

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# Worked example: Completing the square (intro)

Sal completes x²-44x into a perfect square. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- What's the difference between a binomial and trinomial? Also, is there any other vocabulary I should know? :) Thank you(21 votes)
- A binomial consists of 2 terms, hence the math prefix bi which means 2, as in bicycle. In contrast, a trinomial includes 3 terms, the math prefix tri means 3, as in tricycle. Hope that helps(57 votes)

- I don't understand, while I'm doing completing the square, that the second variable I can't half it

Ex : x2-19x= -84(10 votes)- Ruth,

x²-19x= -84

Half of 19 is 19/2 so you get to work with fractions.

19/2 squared is 361/4 so you add 361/4 to both sides.

x²-19x+361/4= -84 + 361/4

(x-19/2)² = -84+361/4 Find a common denominator

(x-19/2)² = -336/4+361/4

(x-19/2)² = -336/4+361/4

(x-19/2)² = 25/4 Take the square root of both sides

x-19/2 = ±√(25/4)

x-19/2 = ±5/2 Add 19/2 to each side

x=19/2±5/2

x=24/2 and x=14/2

x=12 and x=7

Fractions can make it tricky so check your answers

x²-19x= -84

12²-19(12)= -84

144-228 = -84

-84 = -84

7²-19(7) = -84

49-133 = -84

-84 = -84

So both answers check correctly

x=12 and x=7

I hope that helps make it click for you.(44 votes)

- i dont understand why is C= 22(6 votes)
- take -44 halve it, -22, then to get C -22 squared -22 * -22 = 484(27 votes)

- is the graph of a perfect square an upwards parabola with an axis of symmetry?(10 votes)
- Yes, in fact, vertex form uses perfect squares to create parabolas
`y = a(x – h)² + k`

As long as 'a' is positive, the graph of a perfect square will be an upwards parabola symmetric along the line x=h.(19 votes)

- at0:53couldn't you just put ax squared?(4 votes)
- No, because it has a (+) sign in between the two (ax) terms instead of a multiplication sign. Think of it this way, if you have 1x+1x you wouldn't say 1x squared. You would say 2x. There is an invisible 1 in front of the x. Hope this helps you out! :)(5 votes)

- Why is it so important to make the equations perfect squares?(7 votes)
- If I understand your question...it's important to get a perfect square on one side of the equation to be able to take the square root of both sides.(7 votes)

- Is x equals 22 minus (plus or minus the square root of negative 484) even possible?(4 votes)
- No, at least not with this problem. If the problem had been an equation of:

x^2-44x = 0

Completing the square would have resulted in

x^2-44x+484 = 484

(x-22)^2 = 484

Take square root: x-22 = +/- sqrt(484)

Simplify: x = 22 +/- 22

This results in: x=22+22 = 44

And in x = 0

Note: The equation would be easier to solve using factoring.

Hope this helps.(10 votes)

- What do you do if the coefficient is odd and can't be divided by 2 evenly?(4 votes)
- completing the square works only for quadratics, or degree 2 polynomials. OR if you can change another into one. For instance if you had x^4 - 5x^2 + 6 you could set x^2 = y and now you would have y^2 - 5y + 6. Just keep in mind after completing the square you have to put x^2 back in for y.

if the degree is greater than 2 now, and you cannot make it look like a degree 2 you cannot use completing the square. You will have to use other methods you learn later on.(3 votes)

- i dont get that pattern about (x+a) squared is because my teacher showed us a different way(3 votes)
- how did he get from

x^2+2(-22)x+(-22)^2

to

(x-22)^2(3 votes)- Both equations are the same so to save time and space you combine the two because ^2 means to multiply something by itself:)

Hope this helps!(3 votes)

## Video transcript

Use completing the square to
find the value of c that makes x squared minus 44x plus c--
so we can just figure out a c-- that makes it a perfect
square trinomial-- and a trinomial is just a polynomial
with three terms here. Then write the expression as
the square of a binomial. So we have x squared
minus 44x plus c. So how do we make this into
a perfect square? Well, if you just look at the
traditional pattern for a perfect square, let's
just think of it in terms of x plus a squared. That's the same thing as x plus
a times x plus a, and we've seen this before. And if you were to multiply this
out, that's x times x, which is x squared, plus
x times a, which is ax. Plus a times x, which is ax. Plus a times a, which
is a squared. So it's x squared plus 2ax,
these two, you have an ax plus an ax gives you 2ax,
plus a squared. So if we can get this into this
pattern, where I have whatever value is here, if
I take half of it, right? This is going to be 2a here. If I take half of it and square
it over here, then this will be a perfect square. So if we look over here, this
thing right here is 2a, if we want to pattern match, if we
want to make this look like a perfect square. That has to be 2a. So negative 44 is equal to 2a. And this, right here, this c, if
we pattern match, c has to be equal to a squared. So what's a? Well, if we know negative 44
is 2a, we can divide both sides of that by 2. And we know that negative 22
has got to be equal to a. a has got to be equal
to negative 22. a is half of the coefficient
right here. It's half of negative 44. And whenever you complete the
square, it's always going to be half of the coefficient
right here. Now, if that's a, what
does c need to be? Well, c needs to be a squared
in order for this to be a perfect square. So c needs to equal negative
22 squared. And we can figure out
what that is. 22 times 22, we could put the
negative later-- actually it's just going to be the same thing
because the negative times a negative
is a positive. 2 times 22 is 44, put a 0. 2 times 22 is 44. Get a 4, get an 8, get a 4. So it's 484. So if we were to rewrite this
as x squared minus 44x plus 484, then this is a perfect
square trinomial. Or we could write
it like this. This is x squared minus 2
times-- or maybe I should write it this way-- plus 2
times negative 22x plus negative 22 squared. And when you view it that way,
it's pretty clear that this is a perfect square, and if you
were to factor it, it's the same thing as x minus
22 times x minus 22, or x minus 22 squared. These are all equivalent
statements.