Simplify the expression completely if possible.
start fraction, 3, x, squared, plus, 6, x, divided by, x, squared, minus, 64, end fraction
x
2
−64
3x
2
+6x
To simplify the expression $\frac{3x^2+6x}{x^2-64}$, we can factor both the numerator and the denominator:
The numerator can be factored out common factor $3x$:
$3x(x+2)$
The denominator can be factored as a difference of squares:
$x^2-64 = (x+8)(x-8)$
Therefore, the expression can be simplified as:
$\frac{3x(x+2)}{(x+8)(x-8)}$
Note that we cannot simplify the expression any further, as there are no common factors between the numerator and the denominator.
To simplify the expression completely, we start by factoring the numerator and the denominator.
The numerator, 3x^2 + 6x, can be factored as 3x(x + 2).
The denominator, x^2 - 64, is a difference of squares and can be factored as (x - 8)(x + 8).
Therefore, the simplified expression is (3x(x + 2))/(x - 8)(x + 8)
To simplify the given expression, we need to simplify both the numerator and the denominator separately and then combine the results.
Let's start by simplifying the numerator: 3x^2 + 6x.
We notice that both terms have a common factor of x. Factoring out the common factor, we get:
3x^2 + 6x = x(3x + 6)
Next, let's simplify the denominator: x^2 - 64.
We recognize that this is a difference of squares. Factoring the denominator, we get:
x^2 - 64 = (x + 8)(x - 8)
Now, we can simplify the entire expression:
(start fraction) (3x^2 + 6x) ÷ (x^2 - 64) (end fraction) = (x(3x + 6)) ÷ ((x + 8)(x - 8))
To determine if there are any common factors that can be canceled out, we inspect each individual factor in the numerator and the denominator. In this case, none of the factors can be canceled out because they are all different.
Therefore, the expression cannot be further simplified.