Solve for z 3z^2+12z-20=5z

$3 z^{2} + 12 z - 20 = 5 z$

Move all terms containing $z$ to the left side of the equation.

Subtract $5 z$ from both sides of the equation.

$3 z^{2} + 12 z - 20 - 5 z = 0$

Subtract $5 z$ from $12 z$ .

$3 z^{2} + 7 z - 20 = 0$

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 $a \cdot c = 3 \cdot - 20 = - 60$ and whose sum is $b = 7$ .

Factor $7$ out of $7 z$ .

$3 z^{2} + 7 (z) - 20 = 0$

Rewrite $7$ as $- 5$ plus $12$

$3 z^{2} + (- 5 + 12) z - 20 = 0$

Apply the distributive property.

$3 z^{2} - 5 z + 12 z - 20 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 z^{2} - 5 z) + 12 z - 20 = 0$

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

$z (3 z - 5) + 4 (3 z - 5) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 z - 5$ .

$(3 z - 5) (z + 4) = 0$

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

$3 z - 5 = 0$

$z + 4 = 0$

Set $3 z - 5$ equal to $0$ and solve for $z$ .

Set $3 z - 5$ equal to $0$ .

$3 z - 5 = 0$

Solve $3 z - 5 = 0$ for $z$ .

Add $5$ to both sides of the equation.

$3 z = 5$

Divide each term in $3 z = 5$ by $3$ and simplify.

Divide each term in $3 z = 5$ by $3$ .

$\frac{3 z}{3} = \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 z}{3} = \frac{5}{3}$

Divide $z$ by $1$ .

$z = \frac{5}{3}$

Set $z + 4$ equal to $0$ and solve for $z$ .

Set $z + 4$ equal to $0$ .

$z + 4 = 0$

Subtract $4$ from both sides of the equation.

$z = - 4$

The final solution is all the values that make $(3 z - 5) (z + 4) = 0$ true.

$z = \frac{5}{3}, - 4$

The result can be shown in multiple forms.

Exact Form:

$z = \frac{5}{3}, - 4$

Decimal Form:

$z = 1.\overline{6}, - 4$

Mixed Number Form:

$z = 1 \frac{2}{3}, - 4$

Solve for z 3z^2+12z-20=5z

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