2xsquare- 8x - 42 over 6x square divided by x square - 9 over x square - 3 x
To simplify the expression (2x^2 - 8x - 42) / (6x^2) divided by (x^2 - 9) / (x^2 - 3x), follow these steps:
Step 1: Factorize the numerator and denominator of each fraction separately.
Starting with the numerator of the first fraction:
2x^2 - 8x - 42
This can be factored as:
2(x^2 - 4x - 21)
Next, factorize the denominator of the first fraction:
6x^2
This can be simplified as:
6(x^2)
Now, move on to the numerator of the second fraction:
x^2 - 9
This can be factorized using the difference of squares formula:
(x - 3)(x + 3)
Lastly, factorize the denominator of the second fraction:
x^2 - 3x
This can be simplified as:
x(x - 3)
Step 2: Rewrite the expression using the factored form of the fractions:
((2(x^2 - 4x - 21)) / (6(x^2))) / (((x - 3)(x + 3)) / (x(x - 3)))
Step 3: Simplify the expression by canceling out common factors in the numerator and denominator:
Starting with the numerator, cancel out common factors:
2(x^2 - 4x - 21) = 2(x - 7)(x + 3)
Moving on to the denominator, cancel out common factors:
6(x^2) = 6x^2
In the second fraction, cancel out common factors:
(x - 3)(x + 3) = (x - 3)(x + 3)
Also, cancel out common factors:
x(x - 3) = x(x - 3)
The expression simplifies to:
(2(x - 7)(x + 3)) / (6x^2) divided by (x - 3)(x + 3) / x(x - 3)
Step 4: Simplify the division of fractions by multiplying the numerator of the first fraction by the reciprocal of the second fraction:
Dividing fractions is the same as multiplying by the reciprocal, so the expression becomes:
(2(x - 7)(x + 3)) / (6x^2) * (x(x - 3)) / ((x - 3)(x + 3))
Step 5: Simplify further by canceling out common factors:
(x - 3) cancels out in both the numerator and denominator:
(2(x - 7) * x) / (6x^2) * (x) / (1(x + 3))
Simplify further by canceling out common factors:
(x - 7) / (3x)
Thus, the simplified expression is:
(x - 7) / (3x)