2x^2-8x-42 over 6x^2 divided by x^2-9 over x^2-3x
2x^2-8x-42 over 6x^2 divided by x^2-9 over x^2-3x
= (2x^2-8x-42)/(6x^2) ÷ [(x^2-9)/(x^2-3x)]
= 2(x+3)(x-7/(6x^2)*x(x-3)/[(x+3)(x-3)]
= (x-7)/(3x) after some lovely canceling.
(x-3)(x^2+x-7)
To simplify the expression (2x^2 - 8x - 42) / (6x^2) ÷ (x^2 - 9) / (x^2 - 3x), we can use a few steps.
Step 1: Simplify the expression in the numerator of the main fraction
The numerator is given as (2x^2 - 8x - 42). We can factor this quadratic expression as (2x - 14)(x + 3).
Step 2: Simplify the expression in the denominator of the main fraction
The denominator is given as 6x^2. This expression does not factor further.
Step 3: Simplify the expression in the numerator of the divisor fraction
The numerator of the divisor fraction is (x^2 - 9). This expression is a difference of squares and factors into (x + 3)(x - 3).
Step 4: Simplify the expression in the denominator of the divisor fraction
The denominator of the divisor fraction is (x^2 - 3x). This expression can be factored as x(x - 3).
Now, let's rewrite the expression with the simplified forms:
[(2x - 14)(x + 3) / 6x^2] ÷ [(x + 3)(x - 3) / x(x - 3)]
When dividing fractions, we multiply the first fraction by the reciprocal of the second fraction. Therefore, we can rewrite the expression as follows:
[(2x - 14)(x + 3) / 6x^2] * [x(x - 3) / (x + 3)(x - 3)]
Next, let's simplify each term separately before multiplying.
In the first fraction:
- Simplify (2x - 14) / 2 = x - 7
- Simplify (x + 3) / x = 1 + 3/x
- Simplify 6x^2 / (x + 3)(x - 3) by canceling common factors:
= 6x / (x - 3)
In the second fraction:
- Simplify x(x - 3) / x = x - 3
- (x + 3)(x - 3) / (x + 3)(x - 3) simplifies to 1 since the numerator and denominator are equal.
Now, we can rewrite the expression as:
(x - 7) * (1 + 3/x) * (6x / (x - 3)) * (x - 3) * 1
By canceling out common factors and simplifying further, the expression reduces to:
(x - 7) * (3 + 18x) / (x - 3)
Therefore, the simplified expression is:
(3x - 7) * (3 + 18x) / (x - 3)