[(x^2 -16) / (2x^2-9x + 4)] / [(2x^2 + 14x + 24) / (4x + 4)]
is the answer 1 / x + 3
To solve this expression, we can simplify it by following these steps:
Step 1: Factorize the numerator and the denominator separately.
The numerator of the left fraction can be factored as the difference of squares: (x^2 - 16) = (x + 4)(x - 4).
The denominator can be factored as a quadratic equation: (2x^2 - 9x + 4) = (2x - 1)(x - 4).
The numerator of the right fraction can be factored as a quadratic equation: (2x^2 + 14x + 24) = 2(x^2 + 7x + 12) = 2(x + 3)(x + 4).
The denominator can be factored as a linear equation: (4x + 4) = 4(x + 1).
Now our expression becomes:
[(x + 4)(x - 4) / (2x - 1)(x - 4)] / [2(x + 3)(x + 4) / 4(x + 1)]
Step 2: Simplify the expression by canceling out common factors.
In the numerator, (x - 4) cancels out, and in the denominator, (x + 4) cancels out. Additionally, the coefficient 2 in the denominator cancels out with the 4 in the numerator.
Simplifying further, we get:
[(x + 4) / (2x - 1)] / [2(x + 3) / (x + 1)]
Step 3: Simplify further by dividing fractions.
To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
Reciprocal of a fraction is obtained by swapping the numerator and the denominator.
Therefore, our expression becomes:
[(x + 4) / (2x - 1)] * [(x + 1) / 2(x + 3)]
Step 4: Multiply the numerators and the denominators separately.
When multiplying expressions, we multiply the numerators together and the denominators together.
So our expression becomes:
[(x + 4)(x + 1)] / [(2x - 1)(2)(x + 3)]
Step 5: Simplify the expression by expanding and combining like terms.
Expanding the numerator, we get: x^2 + 5x + 4.
Expanding the denominator, we get: 4(2x - 1)(x + 3) = 4(2x^2 + 5x + 1).
Our simplified expression becomes:
(x^2 + 5x + 4) / [4(2x^2 + 5x + 1)]
So, the expression [(x^2 - 16) / (2x^2 - 9x + 4)] / [(2x^2 + 14x + 24) / (4x + 4)] simplifies to (x^2 + 5x + 4) / [4(2x^2 + 5x + 1)].
The answer is not 1 / (x + 3).