Math problem. Help!

Question

The expression 1/3x^2 - 2 can be rewritten as 1/3(x-k)(x+k), where k is a positive constant. What is the value of k? A) 2 B) 6 C) square root of 2 D) square root of 6

Answer 1

Note: it looks like this question was solved differently while I wrote this. If you're not sure why we got different answers, the answer above assumes you meant 1/3(x²-2), and i assume 1/3x²-2.

------------------------------------------------------------------------------------------------------------------------

What this SAT question wants is for us to rearrange 1/3x²-2 to look as much like 1/3(x+k)(x-k) as possible, but with numbers where the "k"s are.

It can be tricky to solve these problems since the new equations have variables that weren't there before. I really like working backwards, rearranging 1/3(x-k)(x+k) to look like our original formula 1/3x²-2.

We can use a difference of squares equation first, which tells us x²-k² is equal to (x+k)(x-k). Then we can say

1/3(x-k)(x+k) = 1/3(x²-k²).

Once we've distributed the 1/3, we get

1/3(x²-k²)=1/3x²-1/3k².

We're rearranging until our equation looks as close as possible to 1/3x²-2, and it looks like we almost made it. Whatever k is, when we plug it in to 1/3k², we should get 2.

In other words, 1/3k²=2. Let's solve this for k.

Multiplying both sides by 3, we get k²=6. Taking the square root of both sides, I see that k is the square root of 6, which is one of our answers. And we're done!

To recap, we

-- worked backwards, rearranging the formula with mystery variables to look like our original formula

-- used a difference of squares equation

-- figured out a good value for k

Answer 2

1/3x^2 means 1/(3x^2). I doubt that you meant for the x^2 to be in the denominator.

Did you mean to write (1/3)(x^2-2)???

x^2 - 2 is the difference of square, like (a^2 - b^2) which factors to (a+b)(a-b)

Now (x^2 - 2) = (x^2 - (sqrt(2))^2) = (x+ sqrt(2))(x - sqrt(2))

Then (1/3)(x^2-2)= (1/3)(x+ sqrt(2))(x - sqrt(2))

So k = ???

Answer 3

x² - 2 is the difference of two squares.

Recall a² - b² = (a-b)(a+b)

Also note that x² - 2 = x² - (sqrt 2)² = (x -sqrt 2)(x+ sqrt 2)

So k = sqrt of 2