x^2-10x+25/3x+12 * x^2-16/2x-10
To simplify the expression (x^2-10x+25)/(3x+12) * (x^2-16)/(2x-10), we can factor the numerator and denominator of each fraction.
(x^2-10x+25) can be factored as (x-5)(x-5) or (x-5)^2.
(3x+12) can be factored as 3(x+4).
(x^2-16) can be factored as (x-4)(x+4).
(2x-10) can be factored as 2(x-5).
Now we can rewrite the expression:
((x-5)(x-5))/(3(x+4)) * ((x-4)(x+4))/(2(x-5))
We can then cancel out common factors:
((x-5)(x-5))/(3(x+4)) * ((x-4)(x+4))/(2(x-5))
= [(x-5)(x-5)(x-4)(x+4)] / [3(x+4)(x-5) * 2(x-5)]
Now we can simplify further:
= [(x-5)(x-4)(x+4)] / [6(x+4)(x-5)]
= (x-4) / 6
Therefore, the simplified expression is (x-4)/6.
find domain of it
The domain of the expression is the set of all possible values that x can take without resulting in undefined or non-real values.
In this case, the only potential issue is division by zero, as dividing by zero is undefined. So we need to find the values of x that make the denominators in the expression equal to zero.
First, let's consider the expression (3x+12). Setting this denominator equal to zero:
3x+12 = 0
3x = -12
x = -4
So x cannot be equal to -4.
Now, let's look at the other denominator, which is (2x-10). Setting this equal to zero:
2x-10 = 0
2x = 10
x = 5
So x cannot be equal to 5.
Therefore, the domain of the expression is all real numbers except -4 and 5. In interval notation, it can be expressed as (-∞, -4) U (-4, 5) U (5, ∞).
both wrong
Apologies for the mistake. Let's find the correct domain of the expression.
To determine the domain, we need to identify any values that would result in division by zero or any other undefined operations.
Let's consider the denominators:
For the first fraction, 3x + 12, division by zero occurs when 3x + 12 = 0. Solving this equation, we find:
3x = -12
x = -4
So x cannot be equal to -4.
For the second fraction, 2x - 10, division by zero occurs when 2x - 10 = 0. Solving this equation, we have:
2x = 10
x = 5
Therefore, x cannot be equal to 5 either.
Thus, the values x = -4 and x = 5 are not in the domain of the expression.
Therefore, the domain of the expression (x^2-10x+25)/(3x+12) * (x^2-16)/(2x-10) is all real numbers except for x equal to -4 and x equal to 5.
In interval notation, the domain can be written as:
(-∞, -4) U (-4, 5) U (5, ∞).
now simplify correctly
Apologies for the previous mistakes. Let's simplify the expression correctly.
The expression is: (x^2-10x+25)/(3x+12) * (x^2-16)/(2x-10)
First, let's factor the numerator and denominator of each fraction:
Numerator of the first fraction, x^2 - 10x + 25, can be factored as (x-5)(x-5) or (x-5)^2.
Denominator of the first fraction, 3x + 12, can be factored as 3(x + 4).
Numerator of the second fraction, x^2 - 16, can be factored as (x-4)(x+4).
Denominator of the second fraction, 2x - 10, can be factored as 2(x - 5).
Now we can rewrite the expression:
((x-5)(x-5))/(3(x+4)) * ((x-4)(x+4))/(2(x-5))
Next, we can cancel out common factors between numerators and denominators:
((x-5)(x-5))/(3(x+4)) * ((x-4)(x+4))/(2(x-5))
= [(x-5)(x-4)(x+4)] / [3(x+4) * 2]
Further simplifying gives:
= [(x-5)(x-4)(x+4)] / [6(x+4)]
Now, we can cancel out the (x+4) terms:
[(x-5)(x-4)(x+4)] / [6(x+4)]
= (x-5)(x-4) / 6
Therefore, the simplified expression is (x-5)(x-4) / 6.
Apologies again for the confusion in the previous response. Let me know if there's anything else I can help you with!
1-1/x/1+1/x
find domain
To find the domain of the expression (1 - 1/x) / (1 + 1/x), we need to identify any values of x that would result in undefined operations.
Division by zero is undefined, so we need to find the values of x that make the denominator of the expression equal to zero.
For the denominator, 1 + 1/x, division by zero occurs when 1 + 1/x = 0. Solving this equation, we have:
1/x = -1
x = -1
So x cannot be equal to -1.
Therefore, the domain of the expression (1 - 1/x) / (1 + 1/x) is all real numbers except for x equal to -1.
In interval notation, we can express the domain as:
(-∞, -1) U (-1, ∞)