math help please please please?

Question

can someone please explain to me how to do this??
9-x2(squared)
----------------------------
10x2(squared)-28x-6

oh sorry it also says "express the following rational expression in simplest form"

Answer 1

(9 - x²)/(10x² - 28x - 6)

Factor the numerator and denominator:
(3 - x)(3 + x)/(10x + 2)(x - 3)

Multiply the (3 - x) by -1 to get (x - 3) and cancel:
-(3 + x)/(10x + 2)

Factor out the 2 in the denominator:
-(3 + x)/(2(5x + 1))

Answer 2

(FYI, use the carat key (^ "shift 6") to show raising to a power)

This is a complex fraction that can only be simplified after you factor both the top(numerator) and bottom(denominator). After these have been simplified, if you have any common terms or binomials, they can be "canceled" from the top and bottom.

The top 9 - x^2 is called the difference of two squares. This is when there is subtraction between two terms, both of which are perfect squares. To factor this, write two sets of parenthesis, one with a + and one with a - sign. The first position in each parenthesis will be filled with the square root of the first term, and the second position is filled with the square root of the second term:

9 - x^2
(3 + x)(3 - x)

You can distribute/multiply to check this factoring.

The denominator is a quadratic equation that needs to first have a greatest common factor taken out to make things easier.

10x^2 - 28x - 6
2(5x^2 - 14x - 3)

Now, this quadratic inside the parenthesis will again break into two binomials multiplying together. I'm assuming that you've had some practice doing this based on the difficulty of this problem.

2(5x^2 - 14x - 3)
2(5x + 1)(x - 3)

Now, let's put the numerator and denominator back together

(9 - x^2) / (10x^2 - 28x - 6)
(3 - x)(3 + x) / 2(5x + 1)(x - 3)

Look how close the first binomial of the numerator 3 - x, and the last binomial of the denominator x - 3 are to matching up! If you factor a -1 out of either one, they will match exactly and cancel out. I'll factor the -1 from the top, and every sign will change. The denominator will not change.

(3 - x)(3 + x) / 2(5x + 1)(x - 3)
-1(-3 + x)(-3 - x) / 2(5x + 1)(x - 3)

The commutative property says we can rewrite that first binomial in the numerator.

-1(x - 3)(-3 - x) / 2(5x + 1)(x - 3)

Now, since there is an (x - 3) in both the top and bottom, they will cancel out and disappear!

-1(-3 - x) / 2(5x + 1)

and you can distribute to get back to simplest form.

(3 + x) / (10x + 2)

or if your prof. is really picky...

(x + 3) / (10x + 2)

Answer 3

your equation is very confusing