Simplify the difference. State any restrictions on the variable.

Question

3x/x^2+x-6 - 6x/x^2-5x+6

Answer 1

To simplify the difference, we need to first find a common denominator. The denominators of the two fractions are (x^2 + x - 6) and (x^2 - 5x + 6).

To find the common denominator, we need to factor both denominators:

x^2 + x - 6 = (x + 3)(x - 2)
x^2 - 5x + 6 = (x - 2)(x - 3)

The common denominator is (x + 3)(x - 2)(x - 3).

Now we can rewrite the fractions with the common denominator:

(3x/(x^2 + x - 6)) - (6x/(x^2 - 5x + 6))

Multiplying the first fraction by (x - 2)/(x - 2) and the second fraction by (x + 3)/(x + 3), we get:

(3x(x - 2)/((x^2 + x - 6)(x - 2))) - (6x(x + 3)/((x^2 - 5x + 6)(x + 3)))

Simplifying further:

(3x^2 - 6x)/((x + 3)(x - 2)(x - 3)) - (6x^2 + 18x)/((x + 3)(x - 2)(x - 3))

Now, we can subtract the two fractions:

(3x^2 - 6x - 6x^2 - 18x)/((x + 3)(x - 2)(x - 3))

Simplifying the numerator:

(-3x^2 - 24x)/((x + 3)(x - 2)(x - 3))

Finally, we can simplify further:

-3x(x + 8)/((x + 3)(x - 2)(x - 3))

There are no restrictions on the variable.

Answer 2

close, but no cigar...

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

Answer 3

Apologies for the mistake. You are correct, the simplified difference should be:

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

Thank you for catching that.