Subtract. x/3x+76 - x/9x+21
To subtract the given expression, x/3x+76 - x/9x+21, we need to find a common denominator for the fractions.
The common denominator can be found by finding the least common multiple (LCM) of the denominators, which are 3x+76 and 9x+21.
The LCM of 3x+76 and 9x+21 can be found by factoring the denominators:
3x+76 can be factored into 3(x+25)
9x+21 can be factored into 3(3x+7)
Therefore, the LCM is 3(x+25)(3x+7).
Now, let's rewrite the fractions with the common denominator:
x/3x+76 becomes (x * 3(3x+7))/(3(x+25)(3x+7))
x/9x+21 becomes (x * (x+25))/(3(x+25)(3x+7))
Now we can subtract the fractions by subtracting the numerators while keeping the common denominator:
(x * 3(3x+7))/(3(x+25)(3x+7)) - (x * (x+25))/(3(x+25)(3x+7))
Combining the numerators, we get:
(3x^2 + 7x - x^2 - 25x)/(3(x+25)(3x+7))
Simplifying the numerator gives us:
(2x^2 - 18x)/(3(x+25)(3x+7))
Therefore, the simplified expression is (2x^2 - 18x)/(3(x+25)(3x+7)).
To subtract the given fractions:
Step 1: Find the least common denominator (LCD) by taking the least common multiple (LCM) of the denominators.
The denominators are 3x+76 and 9x+21.
The factors of 3x+76 are (3, (x+25)).
The factors of 9x+21 are (3, (3x+7)).
The LCM of these two expressions will be the product of all the unique factors, each taken to the maximum power it appears in either expression.
So, the LCM of 3x+76 and 9x+21 is (3) * (x+25) * (3x+7).
Step 2: Rewrite each fraction with the LCD as the common denominator.
(x/3x+76) = (x * (3) * (x+25) * (3x+7)) / ((3x+76) * (x+25) * (3x+7))
(x/9x+21) = (x * (3) * (x+25) * (3x+7)) / ((9x+21) * (x+25) * (3x+7))
Step 3: Perform the subtraction.
[(x * (3) * (x+25) * (3x+7)) / ((3x+76) * (x+25) * (3x+7))] - [(x * (3) * (x+25) * (3x+7)) / ((9x+21) * (x+25) * (3x+7))]
= (3x^3 + 75x^2 + 21x) / ((3x+76) * (x+25) * (3x+7)) - (3x^3 + 75x^2 + 21x) / ((9x+21) * (x+25) * (3x+7))
Step 4: Combine the fractions if possible.
In this example, the terms can cancel each other out:
(3x^3 + 75x^2 + 21x) - (3x^3 + 75x^2 + 21x) = 0
Therefore, the result of the subtraction is 0.
To subtract the given expressions, x/3x+76 - x/9x+21, we need to find a common denominator for the two fractions.
The expressions have denominators of (3x+76) and (9x+21). To find a common denominator, we need to determine the least common multiple (LCM) of (3x+76) and (9x+21).
Step 1: Factorize the denominators
- (3x+76) can be factored further, but it seems difficult to find the exact factors without using algebraic techniques such as completing the square or using the quadratic formula.
- (9x+21) can be factorized as 3(3x+7).
Step 2: Compare the factors of the two denominators
- By comparing the factors of the denominators, we can see that (3x+7) is present in both denominators.
Step 3: Determine the common denominator
- Since (3x+7) is a common factor, we just need to consider the additional factors from each denominator. In this case, (3x+76) has an additional factor of 4.
- Therefore, the common denominator is 4(3x+7) or 12x+28.
Now that we have the common denominator, we can subtract the fractions:
x/3x+76 - x/9x+21 = (x(9x+21) - x(3x+76))/(12x+28)
Simplifying further:
= (9x^2 + 21x - 3x^2 - 76x)/(12x+28)
= (6x^2 - 55x)/(12x+28)
Thus, the simplified expression after subtracting the fractions is (6x^2 - 55x)/(12x+28).