Solve for z 3z^2=-23z-40

$3 z^{2} = - 23 z - 40$

Step 1

Add $23 z$ to both sides of the equation.

$3 z^{2} + 23 z = - 40$

Step 2

Add $40$ to both sides of the equation.

$3 z^{2} + 23 z + 40 = 0$

Step 3

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 $b = 23$ .

Factor $23$ out of $23 z$ .

$3 z^{2} + 23 (z) + 40 = 0$

Rewrite $23$ as $8$ plus $15$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 4

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

$3 z + 8 = 0$

$z + 5 = 0$

Step 5

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 $z$ .

Subtract $8$ from 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 $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 6

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.

$z = - 5$

Step 7

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$z = - 2.\overline{6}, - 5$

Mixed Number Form:

$z = - 2 \frac{2}{3}, - 5$

Solve for z 3z^2=-23z-40

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