Find the Excluded Values (-3z+6)/(6z^2+14z-80)

$\frac{- 3 z + 6}{6 z^{2} + 14 z - 80}$

Set the denominator in $\frac{- 3 z + 6}{6 z^{2} + 14 z - 80}$ equal to $0$ to find where the expression is undefined.

$6 z^{2} + 14 z - 80 = 0$

Solve for $z$ .

Factor the left side of the equation.

Factor $2$ out of $6 z^{2} + 14 z - 80$ .

Factor $2$ out of $6 z^{2}$ .

$2 (3 z^{2}) + 14 z - 80 = 0$

Factor $2$ out of $14 z$ .

$2 (3 z^{2}) + 2 (7 z) - 80 = 0$

Factor $2$ out of $- 80$ .

$2 (3 z^{2}) + 2 (7 z) + 2 (- 40) = 0$

Factor $2$ out of $2 (3 z^{2}) + 2 (7 z)$ .

$2 (3 z^{2} + 7 z) + 2 (- 40) = 0$

Factor $2$ out of $2 (3 z^{2} + 7 z) + 2 (- 40)$ .

$2 (3 z^{2} + 7 z - 40) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 40 = - 120$ and whose sum is $b = 7$ .

Factor $7$ out of $7 z$ .

$2 (3 z^{2} + 7 (z) - 40) = 0$

Rewrite $7$ as $- 8$ plus $15$

$2 (3 z^{2} + (- 8 + 15) z - 40) = 0$

Apply the distributive property.

$2 (3 z^{2} - 8 z + 15 z - 40) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((3 z^{2} - 8 z) + 15 z - 40) = 0$

Factor out the greatest common factor (GCF) from each group.

$2 (z (3 z - 8) + 5 (3 z - 8)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 z - 8$ .

$2 ((3 z - 8) (z + 5)) = 0$

Remove unnecessary parentheses.

$2 (3 z - 8) (z + 5) = 0$

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

$3 z - 8 = 0$

$z + 5 = 0$

Set $3 z - 8$ equal to $0$ and solve for $z$ .

Set $3 z - 8$ equal to $0$ .

$3 z - 8 = 0$

Solve $3 z - 8 = 0$ for $z$ .

Add $8$ to both sides of the equation.

$3 z = 8$

Divide each term in $3 z = 8$ by $3$ and simplify.

Divide each term in $3 z = 8$ by $3$ .

$\frac{3 z}{3} = \frac{8}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 z}{3} = \frac{8}{3}$

Divide $z$ by $1$ .

$z = \frac{8}{3}$

Set $z + 5$ equal to $0$ and solve for $z$ .

Set $z + 5$ equal to $0$ .

$z + 5 = 0$

Subtract $5$ from both sides of the equation.

$z = - 5$

The final solution is all the values that make $2 (3 z - 8) (z + 5) = 0$ true.

$z = \frac{8}{3}, - 5$

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 $0$ .

$z = - 5, z = \frac{8}{3}$

Find the Excluded Values (-3z+6)/(6z^2+14z-80)

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