Simplify the expression, if possible. State the excluded values, if any.
1. 5x - 5/ 10x^2 - 25x + 15
2. w^2 + 6w + 9/ 2w^2 - 18
Thanks so much.
(5x-5)/(10x^2-25x+15)
= (x-1)(2x^2-5x+3)
= (x-1) / (x-1)(2x-3)
= 1/(2x-3)
x=1 and x = 3/2 are excluded because you cannot divide by zero.
(w^2+6w+9)/(2w^2-18)
= (w+3)(w+3) / 2(w+3)(w-3)
= (w+3) / 2(w-3)
x = 3 and -3 are excluded: no division by zero allowed
To simplify the given expressions, we need to factor the numerators and denominators and then cancel out any common factors.
1. 5x - 5 / 10x^2 - 25x + 15
First, let's factor the numerator:
5x - 5 = 5(x - 1)
Now, let's factor the denominator:
10x^2 - 25x + 15 = 5(2x^2 - 5x + 3)
The denominator can be further factored as:
5(2x^2 - 3x - 2x + 3) = 5[(2x^2 - 3x) + (-2x + 3)]
= 5[x(2x - 3) - 1(2x - 3)]
= 5(2x - 3)(x - 1)
Now, we can cancel out the common factors between the numerator and denominator:
5x - 5 / 10x^2 - 25x + 15
= (5(x - 1)) / (5(2x - 3)(x - 1))
The simplified expression is:
(x - 1) / (2x - 3)
In this case, there are no excluded values since there are no denominators that would result in dividing by zero.
2. w^2 + 6w + 9 / 2w^2 - 18
Let's factor the numerator:
w^2 + 6w + 9 = (w + 3)(w + 3) = (w + 3)^2
Now, let's factor the denominator:
2w^2 - 18 = 2(w^2 - 9)
The denominator can be further factored as the difference of squares:
2(w^2 - 9) = 2(w - 3)(w + 3)
Now, we can cancel out the common factors between the numerator and denominator:
(w + 3)^2 / 2(w - 3)(w + 3)
The simplified expression is:
(w + 3) / 2(w - 3)
In this case, there is an excluded value of w = -3, since it would result in dividing by zero (making the expression undefined).