Algebra 3d Ed Page 167 Problem 14 (answer on173). Hard Q

[hrobak.r]

For problem 14, the book indicates that quantity A is larger.

However, I don't understand why the answer is what it is or even why the book chose to "complete the square" to determine the answer. Am I incorrect to simply plug in numbers for "b" to see what "a" is in any given situation? I'm sure that I'm wrong, but I don't see how I'm supposed to understand this problem properly. Help?

[tommywallach]

Hey Hrobak,

I'm with you! You could definitely plug in values, but you'd have to try a lot before you could be really confident, because the numbers are weird. I'd want to try some small numbers, some decimals, and some big numbers:

If b = 0, a = 7
If b = 10, a = four hundred something
If b = -10, a = four hundred something
If b = .5, a = about 4 ish
If b = -.5, a = about 4 ish

At that point, you could confidently say the answer is A. And yes, I do think this is better than coming up with that whole "finish the square" thing. : )

-t

[hrobak.r]

Hey Hrobak,

I'm with you! You could definitely plug in values, but you'd have to try a lot before you could be really confident, because the numbers are weird. I'd want to try some small numbers, some decimals, and some big numbers:

If b = 0, a = 7
If b = 10, a = four hundred something
If b = -10, a = four hundred something
If b = .5, a = about 4 ish
If b = -.5, a = about 4 ish

At that point, you could confidently say the answer is A. And yes, I do think this is better than coming up with that whole "finish the square" thing. : )

-t

Tommy, thanks for your help, but if you wouldn't mind, I still have a couple of questions.

First, why does b=-10 not result in a = 607?
I may be incorrect, but (-10)^2 = 100, which gives us 5(100)-10(-10)+7. This yields 500 + 100 + 7 = 607, correct?

Also, after some study, I understand the concept of completing the square, but I'm having trouble with one of the steps in the answer to the problem in the strategy guide. I follow that we get 5(b-1)^2+2. I also follow that from here we can subtract 2 from each side. However, it does not seem to follow that we could subsequently subtract b from each side of the equation because the original equation was set equal to "a" and not to "b."

For instance, the original equation was 5b^2-10b+7=a. I simply do not understand how in the world the answer book has us subtracting "b" from both sides toward the final steps of the problem unless it mistakenly set the equation equal to "b" instead of "a" from the beginning. The answer explanation seems to set the equation equal to "b" while the question in the problem set has it equal to "a." Does that make sense, or am I missing something?

[tommywallach]

Hey Hrobak,

As for the plugging in of numbers, I was just showing that the one side will always be bigger; as you can see, I didn't really concern myself with the actual arithmetic (it doesn't matter if it's 400 or 500 or 600, because it's still way bigger than 10). Sorry if that added to your confusion.

To your other issue, you're mistaking it a bit. The column that says a is being substituted with the thing that we know equals a, namely the main equation of the question. The column that says b simply stays the way it is. No mistake has been made.

Make sense?

-t

[hrobak.r]

Hey Hrobak,

As for the plugging in of numbers, I was just showing that the one side will always be bigger; as you can see, I didn't really concern myself with the actual arithmetic (it doesn't matter if it's 400 or 500 or 600, because it's still way bigger than 10). Sorry if that added to your confusion.

To your other issue, you're mistaking it a bit. The column that says a is being substituted with the thing that we know equals a, namely the main equation of the question. The column that says b simply stays the way it is. No mistake has been made.

Make sense?

-t

That mostly makes sense. I guess I'm not understanding the way the answer is structured. It says "It's clearer to see the answer if you subtract b and 2 from both sides next." And then it appears to eliminate the b that was previously in the right hand column, as well as the 2 that previously was in the left and column (i.e. in the equation), and subtract the 2 from the column where the b was and subtract the b from the equation in the left column. That appears to me to indicated that the answer key set the equation (left hand column) equal to "b" (right hand column) from the beginning, or else the algebra would not have been done/explained that way.

[tommywallach]

Hey Hrobak,

The original equation is:

5b^2-10b+7=a

Once we replace a with that equation, our two columns are:

Column A: 5b^2 - 10b + 7
Column B: b

Then they manipulate column A a bit, then they subtract b from both columns, then they subtract 2 from both columns. That's why column B becomes -2.

Column B: b - b - 2 --> Column B: -2

Make sense?

-t

[hrobak.r]

Yes, I got it now. Thank you for your help (and your patience).

[tommywallach]

No problem! It's why I'm here! : )

-t