Please simplify this expression,if possible. State the excluded values,if any.
5x-5 / 10x^2 -25x + 15.
thank you for your time.
Divide numerator and denominator by 5:
(x-1)/(2x^2-5x+3).
Factor the denominator:
A*C = 2*3 = 6 = (-2)*(-3),
2x^2 - (2x+3x) + 3 = 0,
(2x^2-2x) - (3x-3) = 0,
2x(x-1) - 3(x-1) = 0,
(x-1)(2x-3) = 0,
(x-1)/(x-1)(2x-3) = 1/(2x-3).
To simplify the given expression, we need to factorize the numerator and the denominator, if possible. Then, we can cancel out any common factors.
Let's start with the numerator:
5x - 5
We notice that both terms have a common factor of 5. So, we can factor it out:
5(x - 1)
Now, let's factorize the denominator:
10x^2 - 25x + 15
First, let's find the greatest common factor (GCF) among the coefficients:
GCF of 10, -25, and 15 is 5
Now, divide each term by 5:
2x^2 - 5x + 3
Next, we try to factorize this quadratic expression. In this case, it's not easily factorable.
Now, we can rewrite the simplified expression:
(5(x - 1)) / (2x^2 - 5x + 3)
There are no excluded values in this expression. However, we can check if there are any values of x that make the denominator equal to zero, as division by zero is undefined. To find these values, we can solve the equation 2x^2 - 5x + 3 = 0 using factoring, quadratic formula, or completing the square methods.
In this case, factoring won't work. So, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = -5, and c = 3. Plugging these values into the formula, we get:
x = (-(-5) ± √((-5)^2 - 4(2)(3))) / (2(2))
x = (5 ± √(25 - 24)) / 4
x = (5 ± √1) / 4
x = (5 ± 1) / 4
Therefore, the excluded values for x are x = 1 and x = 6/4, which simplifies to x = 3/2.
So, the simplified expression is (5(x - 1)) / (2x^2 - 5x + 3), and the excluded values are x = 1 and x = 3/2.