2. Simplify the expression: (x + 3) / (x^2 + 4x - 21) - (x + 1) / (2x - 6)
a. (-x^2 - 6x -1) / (2x^2 + 8x - 42)
b. (-x^2 + 10x + 13) / (2x^2 + 8x - 42)
c. (-x^2 - 6x + 13) / (2x^2 + 8x - 42)
d. (x^2 - 26x - 7) / (2x^2 + 8x - 42)
(x+3) / (x^2 + 4x - 21) - (x+1) / (2x-6)
(x+3) / (x+7)(x-3) - (x+1) / 2(x-3)
now use a common denominator, and you have
2(x+3) / 2(x+7)(x-3) - (x+1)(x+7) / 2(x-3)(x+7)
(2(x+3) - (x+1)(x+7)) / 2(x-3)(x+7)
(-x^2-6x-1) / (2x^2+8x-42)
Hey there! Let's simplify the expression. Are you ready for some mathematical clownery? Here we go!
The expression is looking quite complicated, so let's break it down step by step.
First, let's find a common denominator for the fractions. The denominators in this case are (x^2 + 4x - 21) and (2x - 6).
The common denominator we can use is (x^2 + 4x - 21)(2x - 6).
Now, let's rewrite the fractions with the common denominator:
(x + 3)(2x - 6) / (x^2 + 4x - 21)(2x - 6) - (x + 1)(x^2 + 4x - 21) / (x^2 + 4x - 21)(2x - 6)
Are you following along, or did I lose you in the clown parade?
Now, let's simplify the numerators:
(2x^2 - 6x + 6x - 18 - x^3 - 4x^2 + 21x - x - 104) / (x^2 + 4x - 21)(2x - 6)
Combining like terms:
(-x^3 - 4x^2 + 21x - x - 122) / (x^2 + 4x - 21)(2x - 6)
And voila! We have simplified the expression. So, my friend, the correct answer is (drumroll, please) option C: (-x^2 - 6x + 13) / (2x^2 + 8x - 42).
I hope you enjoyed this math-infused circus act! If you have any more questions, feel free to ask.
To simplify the expression, we need to find a common denominator for the two fractions.
The common denominator for the fractions is the least common multiple of the denominators, which is (x^2 + 4x - 21)(2x - 6).
Now, let's simplify each fraction separately:
The first fraction ((x + 3) / (x^2 + 4x - 21)) can be rewritten as ((x + 3) / [(x + 7)(x - 3)]).
The second fraction ((x + 1) / (2x - 6)) can be rewritten as ((x + 1) / 2(x - 3)).
Now, we can write the original expression with the common denominator:
[(x + 3) / [(x + 7)(x - 3)]] - [(x + 1) / 2(x - 3)]
To combine the fractions, we subtract the numerators and write them over the common denominator:
[(x + 3)(2(x - 3)) - (x + 1)(x + 7)] / [(x + 7)(x - 3)(2(x - 3))]
Expanding the numerators, we get:
[2(x^2 - 3x + 3x - 9) - (x^2 + 8x + 7x + 7)] / [(x + 7)(x - 3)(2x - 6)]
Simplifying further:
[2(x^2 - 6) - (x^2 + 15x + 7)] / [(x + 7)(x - 3)(2x - 6)]
Simplifying the numerator:
[2x^2 - 12 - x^2 - 15x - 7] / [(x + 7)(x - 3)(2x - 6)]
Combine like terms:
[x^2 - 15x - 19] / [(x + 7)(x - 3)(2x - 6)]
At this point, we can't further simplify the expression, so the answer is:
[x^2 - 15x - 19] / [(x + 7)(x - 3)(2x - 6)]
Therefore, the correct answer is d. (x^2 - 15x - 19) / (2x^2 + 8x - 42).
To simplify the given expression, we need to find a common denominator for the two fractions and then combine them.
The denominators are (x^2 + 4x - 21) and (2x - 6).
To find the least common denominator (LCD), we factorize the denominators and take the product of all distinct factors.
The factored form of (x^2 + 4x - 21) is (x + 7)(x - 3).
The factorized form of (2x - 6) is 2(x - 3).
Taking the product of all distinct factors, we get the LCD as 2(x + 7)(x - 3).
Now, we rewrite the fractions using the LCD.
(x + 3) / (x^2 + 4x - 21) - (x + 1) / (2x - 6) becomes [(x + 3)(2(x - 3))] / [2(x + 7)(x - 3)] - [(x + 1)(x + 7)] / [2(x + 7)(x - 3)].
Next, we combine the fractions by subtracting the numerators since they have the same denominator.
[(x + 3)(2(x - 3)) - (x + 1)(x + 7)] / [2(x + 7)(x - 3)].
Expanding the expressions in the numerators, we have:
[2x^2 - 6x + 6x - 18 - (x^2 + 7x + x + 7)] / [2(x + 7)(x - 3)].
Simplifying the numerator further by combining like terms, we get:
[2x^2 - x^2 - 13x - 25] / [2(x + 7)(x - 3)].
Finally, simplifying the expression:
[-x^2 - 13x - 25] / [2(x + 7)(x - 3)].
Therefore, the correct option is a. (-x^2 - 6x - 1) / (2x^2 + 8x - 42).