Simplify. Identify any x-values for which the expression is undefined.

1. (4x-8)/(x^2-2x)

2. (8x-4)/(2x^2+9x-5)

1.

4 x - 8 = 4 ( x - 2 )

x ^ 2 - 2 x = x ( x - 2 )

(4 x - 8 ) / ( x ^ 2 - 2 x ) =

4 ( x - 2 ) / [ x ( x - 2 ) ] =

4 / x

2.

( 8 x - 4 ) = 4 ( 2 x - 1 )

2 x ^ 2 + 9 x - 5 =

2 x ^ 2 + 9 x - 5 + x - x =

( 2 x ^ 2 - x ) + ( 10 x - 5 ) =

x ( 2 x - 1 ) + 5 ( 2 x - 1 ) =

( 2 x - 1 ) ( x + 5 )

( 8 x - 4 ) / ( 2 x ^ 2 + 9 x -5 ) =

4 ( 2 x - 1 ) / [( 2 x - 1 ) ( x + 5 ) ] =

4 / ( x + 5 ) =

To identify any x-values for which the expression is undefined, we need to find the values of x that would make the denominator equal to zero, because division by zero is undefined.

1. (4x-8)/(x^2-2x):
Setting the denominator equal to zero:
x^2 - 2x = 0

Factoring out x:
x(x - 2) = 0

Setting each factor equal to zero gives us two possible solutions:
x = 0
x - 2 = 0

Solving for x gives us:
x = 0
x = 2

Therefore, the expression is undefined at x = 0 and x = 2.

2. (8x-4)/(2x^2+9x-5):
Setting the denominator equal to zero:
2x^2 + 9x - 5 = 0

Using factoring or the quadratic formula, we find that the two solutions for x are:
x = -5/2
x = 1/2

Therefore, the expression is undefined at x = -5/2 and x = 1/2.

To simplify and identify any x-values for which the expressions are undefined, we need to look for any factors in the expressions that could result in a denominator of zero.

1. For the expression (4x-8)/(x^2-2x), the denominator is x^2-2x. To find the values that make the denominator zero, we set x^2-2x equal to zero and solve for x:

x^2 - 2x = 0
x(x - 2) = 0

So either x = 0 or x - 2 = 0. Solving for x, we find that x = 0 or x = 2. These are the x-values for which the expression is undefined.

2. For the expression (8x-4)/(2x^2+9x-5), the denominator is 2x^2+9x-5. To find the values that make the denominator zero, we set 2x^2+9x-5 equal to zero and solve for x:

2x^2 + 9x - 5 = 0

Since the quadratic equation is not easily factorable, we can use the quadratic formula to find the roots:

x = (-9 ± √(9^2 - 4 * 2 * -5)) / (2 * 2)
x = (-9 ± √(81 + 40)) / 4
x = (-9 ± √121) / 4
x = (-9 ± 11) / 4

This gives us two possible values for x: x = (-9 + 11) / 4 = 1/2 or x = (-9 - 11) / 4 = -5/2. These are the x-values for which the expression is undefined.

In summary, for the expression (4x-8)/(x^2-2x), the values 0 and 2 make the expression undefined, and for the expression (8x-4)/(2x^2+9x-5), the values 1/2 and -5/2 make the expression undefined.