find domain
\frac{b^{\:3}+8}{18b^2+27b}\cdot \frac{2b+3}{b^2-2b+4}
To find the domain of the given expression, we need to find the values of the variable that make the expression undefined.
In this case, the expression will be undefined if the denominator of either fraction equals zero. Therefore, we set the denominators equal to zero and solve for the variable:
For the first fraction:
18b^2 + 27b = 0
Factoring out 9b from the expression:
9b(2b + 3) = 0
Setting each factor equal to zero:
9b = 0 or 2b + 3 = 0
b = 0 or 2b = -3
b = -3/2
For the second fraction:
b^2 - 2b + 4 = 0
This quadratic equation does not factor easily, so we can use the quadratic formula to find the solutions:
b = (-(-2) ± √((-2)^2 - 4(1)(4))) / (2(1))
b = (2 ± √(4 - 16)) / 2
b = (2 ± √(-12)) / 2
The expression is undefined for values of b that make the square root of a negative number, which implies that the expression does not have a real domain.
In conclusion, the domain of the given expression is all real numbers except for b = 0, b = -3/2, and the solutions to the quadratic equation b^2 - 2b + 4 = 0, which do not exist as they involve imaginary numbers.
find domain
\frac{y^2-9}{y^2-6y+9}
To determine the domain of the given expression, we need to find the values of y that make the expression undefined.
In this case, the expression will be undefined if the denominator of the fraction equals zero. Therefore, we set the denominator equal to zero and solve for the variable:
y^2 - 6y + 9 = 0
This equation can be factored as (y - 3)(y - 3) = 0, which implies that y = 3.
The expression is undefined when the denominator equals zero, so the domain of the expression is all real numbers except for y = 3.
find domain
\left(\frac{4a}{2-a}-a\right)\div \frac{a+2}{a-2}
To find the domain of the given expression, we need to determine the values of a that make the expression undefined.
For the given expression, the expression as a whole will be undefined if the denominator in the first fraction (\frac{4a}{2-a}-a) or the second fraction (\frac{a+2}{a-2}) equals zero.
Setting the denominator of the first fraction equal to zero:
2 - a = 0
a = 2
Setting the denominator of the second fraction equal to zero:
a - 2 = 0
a = 2
Therefore, for the expression to be undefined, a must equal 2.
The domain of the expression is all real numbers except for a = 2.
wrong
Apologies for the incorrect response.
To find the domain of the given expression, we need to consider any values of 'a' that would result in division by zero or any other invalid mathematical operation.
Starting with the denominators of the fractions involved:
For the first fraction, the denominator is (2 - a). This fraction would be undefined if (2 - a) equals zero. Solving the equation 2 - a = 0 yields a = 2.
For the second fraction, the denominator is (a - 2). This fraction would be undefined if (a - 2) equals zero. Solving the equation a - 2 = 0 yields a = 2.
So, both fractions would be undefined when a = 2.
Now considering the entire expression, we have a division of one fraction by another. For division, the denominator cannot be equal to zero. In this case, the denominator is the second fraction, which is (a + 2)/(a - 2). This fraction would be undefined if (a - 2) equals zero. Solving the equation a - 2 = 0 yields a = 2.
Therefore, the expression as a whole will be undefined when a = 2.
To find the domain of the expression, we state that the expression is defined for all real numbers except a = 2.
Thus, the domain of the given expression is all real numbers except a = 2.
noo its not a=2
Apologies for the mistake.
To find the domain of the given expression, we need to consider any values of 'a' that would result in division by zero or any other invalid mathematical operation.
Starting with the denominators of the fractions involved:
For the fraction \frac{4a}{2-a}, the denominator is (2 - a). This fraction would be undefined if (2 - a) equals zero. Solving the equation 2 - a = 0 yields a = 2.
However, this value of a = 2 should not be included in the domain since it would cause this fraction to be undefined. Therefore, a = 2 is not in the domain.
For the fraction \frac{a+2}{a-2}, the denominator is (a - 2). This fraction would be undefined if (a - 2) equals zero. Solving the equation a - 2 = 0 yields a = 2.
Again, this value of a = 2 should not be included in the domain since it would cause this fraction to be undefined. Therefore, a = 2 is not in the domain.
Considering the entire expression, we have a division of one fraction by another. For division, the denominator cannot be equal to zero. This means that we need to exclude any values of a that make the denominator (a + 2)/(a - 2) equal to zero.
Setting the denominator (a - 2) equal to zero, we find a = 2.
Therefore, a = 2 should not be included in the domain of the expression.
To summarize, the domain of the given expression is all real numbers except a = 2.