Find the Excluded Values ((-x^2+x+20)/(5x^2-25x))÷((x+4)/(2x-14))

$\frac{- x^{2} + x + 20}{5 x^{2} - 25 x} \div \frac{x + 4}{2 x - 14}$

Step 1

Set the denominator in $\frac{- x^{2} + x + 20}{5 x^{2} - 25 x}$ equal to $0$ to find where the expression is undefined.

$5 x^{2} - 25 x = 0$

Step 2

Solve for $x$ .

Factor $5 x$ out of $5 x^{2} - 25 x$ .

Factor $5 x$ out of $5 x^{2}$ .

$5 x (x) - 25 x = 0$

Factor $5 x$ out of $- 25 x$ .

$5 x (x) + 5 x (- 5) = 0$

Factor $5 x$ out of $5 x (x) + 5 x (- 5)$ .

$5 x (x - 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 5 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $5 x (x - 5) = 0$ true.

$x = 0, 5$

Step 3

Set the denominator in $\frac{x + 4}{2 x - 14}$ equal to $0$ to find where the expression is undefined.

$2 x - 14 = 0$

Step 4

Solve for $x$ .

Add $14$ to both sides of the equation.

$2 x = 14$

Divide each term in $2 x = 14$ by $2$ and simplify.

Divide each term in $2 x = 14$ by $2$ .

$\frac{2 x}{2} = \frac{14}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{14}{2}$

Divide $x$ by $1$ .

$x = \frac{14}{2}$

Simplify the right side.

Divide $14$ by $2$ .

$x = 7$

Step 5

Set the denominator in $\frac{\frac{- x^{2} + x + 20}{5 x^{2} - 25 x}}{\frac{x + 4}{2 x - 14}}$ equal to $0$ to find where the expression is undefined.

$\frac{x + 4}{2 x - 14} = 0$

Step 6

Solve for $x$ .

Set the numerator equal to zero.

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 4, x = 0, x = 5, x = 7$

Step 8

Find the Excluded Values ((-x^2+x+20)/(5x^2-25x))÷((x+4)/(2x-14))

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