find domain
a^2+2ac+c^2/a^2+ac-ax-cx
The domain of a function refers to the set of all possible input values for the function. In the given expression:
(a^2 + 2ac + c^2) / (a^2 + ac - ax - cx)
We need to consider the values that satisfy the denominator being non-zero, as division by zero is undefined.
Let's simplify the expression further:
(a^2 + 2ac + c^2) / [(a^2 + ac) - (ax + cx)]
(a^2 + 2ac + c^2) / [a(a + c) - x(a + c)]
(a^2 + 2ac + c^2) / [(a + c)(a - x)]
Now, for the denominator (a + c)(a - x) to be non-zero, both factors need to be non-zero. Therefore, we have two cases:
1. (a + c) ≠ 0:
This means that a and c cannot be equal to -c simultaneously, which implies a ≠ -c.
2. (a - x) ≠ 0:
This implies a ≠ x.
Therefore, the domain of the given expression is all real numbers except a = -c and a = x.
a^2+2ac+c^2/a^2+ac-ax-cx
simplify
The given expression:
(a^2 + 2ac + c^2) / (a^2 + ac - ax - cx)
can be simplified by factoring the numerator and the denominator separately.
Numerator:
(a + c)^2
Denominator:
a^2 + ac - ax - cx
Now, we can rewrite the expression as:
(a + c)^2 / (a(a + c) - x(a + c))
Next, we can factor out the common term (a + c) from both the numerator and the denominator:
[(a + c)(a + c)] / [(a - x)(a + c)]
Finally, we can cancel out the common factor of (a + c) from the numerator and the denominator:
(a + c) / (a - x)
Therefore, the simplified form of the expression is (a + c) / (a - x).
To find the domain of the expression (a^2 + 2ac + c^2) / (a^2 + ac - ax - cx), we need to determine the values of 'a' and 'c' for which the expression is defined.
First, let's look at the denominator: a^2 + ac - ax - cx.
Since a^2 is always non-negative (positive or zero), a^2 + ac - ax - cx is defined as long as the expression does not result in a division by zero.
To avoid division by zero, we need to make sure that the denominator is not equal to zero. So we set it not equal to zero and solve for 'c':
a^2 + ac - ax - cx ≠ 0
Subtracting ac from both sides of the equation:
a^2 - ax - cx ≠ -ac
Factoring out 'c' from the last two terms:
a^2 - (a + x)c ≠ -ac
Dividing both sides of the equation by (a + x):
a^2 - (a + x)c / (a + x) ≠ -ac / (a + x)
Simplifying further:
a^2 / (a + x) - c ≠ -ac / (a + x)
Now, multiplying both sides of the equation by (a + x):
a^2 - c(a + x) ≠ -ac
Expanding the left side of the equation:
a^2 - ac - cx ≠ -ac
Adding ac to both sides:
a^2 - cx ≠ 0
Now, let's consider the numerator: a^2 + 2ac + c^2.
Since this is a polynomial expression, it is defined for all real values of 'a' and 'c'.
Putting everything together, the domain of the expression (a^2 + 2ac + c^2) / (a^2 + ac - ax - cx) is all real values of 'a' and 'c' which satisfy the condition a^2 - cx ≠ 0.