Another Difference of Squares Problem

[Jason76]

Does this logic look ok?

\(\displaystyle x^{2} - 9\)

\(\displaystyle \sqrt[2]{x^{2}} - \sqrt[2]{9}\)

\(\displaystyle x^{2/2} - 3\) - This step could be left out, cause we can see the fractional exponent from the 2nd root. We know the a fractional exponent with an identical numerator and denominator reduces to 1.

\(\displaystyle x - 3\) - Note that the 2nd root of \(\displaystyle x^{2}\) became \(\displaystyle x\)

\(\displaystyle (x)^{2} - (3)^{2}\) Add the 2nd power to both sides.

\(\displaystyle (x - 3)(x + 3)\) Factor out.

 

[mmm4444bot]

You already know that x^2 is the square of x.

You already know that 9 is the square of 3.

You don't need the rigmarole above, to see x^2 - 9 as one square subtracted from another.

Seems clear to me (having read eight months of your posts) that you already understand this, so I would like to know the real reason for this thread. Are you trying to teach others? :cool:

 

[Jason76]

You already know that x^2 is the square of x.

You already know that 9 is the square of 3.

You don't need the rigmarole above, to see x^2 - 9 as one square subtracted from another.

Seems clear to me (having read eight months of your posts) that you already understand this, so I would like to know the real reason for this thread. Are you trying to teach others? :cool:

Just trying a different way to understand it. So if somebody comes up with a more difficult problem, then it will be easier to solve. Also, understanding the mechanics is always good for an overall understanding.

 

[lookagain]

Does this logic look ok?

\(\displaystyle x^{2} - 9\)

\(\displaystyle \sqrt[2]{x^{2}} - \sqrt[2]{9}\) \(\displaystyle \ \ \ \ \) <----- This equals |x| - 3, not x - 3.

\(\displaystyle x^{2/2} - 3\) - This step could be left out, cause we can see the fractional exponent from the 2nd root. We know the a fractional exponent with an identical numerator and denominator reduces to 1.

\(\displaystyle x - 3\) - Note that the 2nd root of \(\displaystyle x^{2}\) became \(\displaystyle x \ \ \ \ \) <----- False.

\(\displaystyle \sqrt{x^2} \ = \ |x|, \ \ not \ \ x.\) \(\displaystyle \ \ \ \) It is true, though, that \(\displaystyle \ (\sqrt{x})^2 \ = \ x.\)


Edit: Some text that was included before was deleted.

 

[mmm4444bot]

if somebody comes up with a more difficult problem, then it will be easier to solve

Factor the following.

(4x-1)^2 - 64

 

[HallsofIvy]

Does this logic look ok?

\(\displaystyle x^{2} - 9\)

\(\displaystyle \sqrt[2]{x^{2}} - \sqrt[2]{9}\)

\(\displaystyle x^{2/2} - 3\) - This step could be left out, cause we can see the fractional exponent from the 2nd root. We know the a fractional exponent with an identical numerator and denominator reduces to 1.

\(\displaystyle x - 3\) - Note that the 2nd root of \(\displaystyle x^{2}\) became \(\displaystyle x\)

\(\displaystyle (x)^{2} - (3)^{2}\) Add the 2nd power to both sides.

1) This is not an equation so you don't have two "sides".
2) There is NO mathematical operation that "adds" a power.

\(\displaystyle (x - 3)(x + 3)\) Factor out.

I think the point of all of that was just to say that \(\displaystyle x^2- 9= x^2- 3^2\). So just say that!

 

[Jason76]

1) This is not an equation so you don't have two "sides".
2) There is NO mathematical operation that "adds" a power.


I think the point of all of that was just to say that \(\displaystyle x^2- 9= x^2- 3^2\). So just say that!

2) There is NO mathematical operation that "adds" a power.

Good point, but you still come up with the right answer.

 

[lookagain]

Good point, but you still come up with the right answer.

/\/\That's an inappropriate comeback. When you write a "Good point, but..." type statement, you are negating the correction to you with the word "but" in this context. What you did is fudging, and the excuse-making for it continues to make it wrong.

 

[Jason76]

/\/\That's an inappropriate comeback. When you write a "Good point, but..." type statement, you are negating the correction to you with the word "but" in this context. What you did is fudging, and the excuse-making for it continues to make it wrong.

The right way would be just to do it in your head. Perhaps you could resort to taking square roots, if you couldn't do that. But you wouldn't write that down and send to the teacher.

 

[mmm4444bot]

just to do it in your head.

Thank you.

Likewise, we may do the extra exercise that I posted earlier in our head:

Factor (4x-1)^2 - 64

We recognize this as a difference of two squares. (On the left 4x-1 is squared, and on the right 8 is squared.)

Simply knowing the special factoring pattern for a difference of squares allows us to immediately realize the following.

(4x-1+8)(4x-1-8)

Therefore, in our head, the factorization of (4x-1)^2-64 simplifies to (4x+7)(4x-9).

Do more difference-of-squares exercises, and you will begin to see this in your head! :cool: