Perfect Square Questions Part 2

Question

1. Explain why it makes sense to call the expression x^2 + 20x + 100 a perfect square.
2. Solve x^2 + 20x + 100 = 81

Please help, I do not understand. Your assistance is very appreciated!

Answer 1

Hi Cameron,

Hope this can help!

So the first thing to ask is what is it looking for? Well the first question asks what that expression is and why it is a perfect square, so solving that will probably help with the second question.

Well to put it simply a square number is just a number that can be written as (something)². so 15 isn't a square number, but 16 is because it can be written as 4².

So how can we take that top expression and write it as (something)². Well based on our rules of factoring we know a quadratic will have two factors and can be written as (x+a)(x+b). if we get lucky and find that a = b then we can write it as (x+a)(x+a) = (x+a)² and that happens to follow that rule right? lets try it with another example (that I happen to know is a perfect square) and try.

x²+4x+4

So by the rules of factoring, if we think it is written as (x+a)(x+b), then we know that factored out is

x² + (a+b)*x + a*b

so what two numbers make these equations equal? well it is the case where:

a + b = 4

a*b = 4

If you do some algebra we find that both a = b = 2! This is perfect because that means we can write this as

x^2 + 4x + 4 = (x+2)(x+2) = (x+2)².

And you can plug any number into x and see that both sides of this equation are equal, which is pretty cool if you ask me that this works for any number you can think of. I'll leave it to you to find if a = b in your case, and if so what is that number? (Hint, it is)

Once you have that fact, you can leverage it to help you solve problem 2. How would you solve:

(x+a)²=81

I'll leave that to you (Hint, there should be two solutions).

Hopefully that gives you a good place to start to solve this on your own. Let me know if you have any more questions!

Answer 2

A perfect square can be written as (something)² so, in the case of the perfect square 64, it is (8)² so "8" is the "something". In the case of x² + 20x + 100, it is (x + 10)² so "x + 10" is the "something".

To solve x² + 20x + 100 = 81 (using the perfect square method), we can first write x² + 20x + 100 as a square so:

(x + 10)² = 81

Now, we can take the square root of both sides of the equation:

√[(x + 10)²] = √81

The square root of a square is just the "something" so:

x + 10 = √81

In taking the square root of 81, obviously 9 is an answer. However, we must not forget that (-9) • (-9) is also 81 so √81 is either 9 or -9. We can write that in shortened form as ±9 so:

x + 10 = ±9

Subtracting 10 from both sides gives us:

x = -10±9

For the "+9" root, we have -10 + 9 which is -1 so x = -1. For the -9 root, we have -10 - 9 = -19 so x = -19

So x = -1 and x = -19

Answer 3

A perfect square will factor so that both of the factors are the same.

x2+20x+100=

(x+10)(x+10)=

(x+10)²

x² + 20x + 100 = 81 First, write is as a perfect square.

(x+10)²=81 Then, take the square root of both side

√(x+10)²=±√81

x+10=9 and x+10=-9 Subtract 10 from both sides of the equations to get 2 solutions.

x=-1 and x=-19

Answer 4

by this they mean that x^2+20x+100=(x+10)^2

{binomial theorem, (a+b)^2=a^2+2ab+b^2 } or just FOIL it out to see that's true.

Then you can solve x^2 + 20x + 100 = 81 by setting

x^2+20x+100=(x+10)^2=81

so

(x+10)^2=81 and take the square root of both sides

x+10=+_9 or

x+10=9 and x+10=-9

so x=-1 or x=-19

So they mean that the quadratic can be perfectly made into a square