what is the excluded value for fraction x +11 over 3x^@ +5x - 2
To find the excluded value of a fraction, we need to determine the values of the variable for which the denominator becomes zero. In this case, the denominator is 3x^2 + 5x - 2.
To find the excluded value, we can set the denominator equal to zero and solve for x.
Setting the denominator to zero, we have:
3x^2 + 5x - 2 = 0
To solve this quadratic equation, we can use factoring, completing the square, or using the quadratic formula.
Let's use factoring to solve it.
First, we need to find two numbers that multiply to give -2 (the constant term) and add up to 5 (the coefficient of the middle term).
The numbers that satisfy these conditions are 2 and -1, because 2 * -1 = -2 and 2 + (-1) = 1.
Next, we rewrite the quadratic equation using these numbers:
3x^2 + 2x - x - 2 = 0
We group the terms:
(3x^2 + 2x) + (-x - 2) = 0
Now, we factor by grouping:
x(3x + 2) - 1(3x + 2) = 0
We can factor out the common binomial (3x + 2):
(3x + 2)(x - 1) = 0
Now, we can set each factor equal to zero and solve for x:
Case 1: 3x + 2 = 0:
3x = -2
x = -2/3
Case 2: x - 1 = 0:
x = 1
So, the excluded values of the fraction are x = -2/3 and x = 1, since these values would make the denominator zero.