Solve for a 18a^2-30=-33a

$18 a^{2} - 30 = - 33 a$

Add $33 a$ to both sides of the equation.

$18 a^{2} - 30 + 33 a = 0$

Factor the left side of the equation.

Factor $3$ out of $18 a^{2} - 30 + 33 a$ .

Factor $3$ out of $18 a^{2}$ .

$3 (6 a^{2}) - 30 + 33 a = 0$

Factor $3$ out of $- 30$ .

$3 (6 a^{2}) + 3 (- 10) + 33 a = 0$

Factor $3$ out of $33 a$ .

$3 (6 a^{2}) + 3 (- 10) + 3 (11 a) = 0$

Factor $3$ out of $3 (6 a^{2}) + 3 (- 10)$ .

$3 (6 a^{2} - 10) + 3 (11 a) = 0$

Factor $3$ out of $3 (6 a^{2} - 10) + 3 (11 a)$ .

$3 (6 a^{2} - 10 + 11 a) = 0$

Factor.

Factor by grouping.

Reorder terms.

$3 (6 a^{2} + 11 a - 10) = 0$

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 $a \cdot c = 6 \cdot - 10 = - 60$ and whose sum is $b = 11$ .

Factor $11$ out of $11 a$ .

$3 (6 a^{2} + 11 (a) - 10) = 0$

Rewrite $11$ as $- 4$ plus $15$

$3 (6 a^{2} + (- 4 + 15) a - 10) = 0$

Apply the distributive property.

$3 (6 a^{2} - 4 a + 15 a - 10) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$3 ((6 a^{2} - 4 a) + 15 a - 10) = 0$

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

$3 (2 a (3 a - 2) + 5 (3 a - 2)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 a - 2$ .

$3 ((3 a - 2) (2 a + 5)) = 0$

Remove unnecessary parentheses.

$3 (3 a - 2) (2 a + 5) = 0$

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

$3 a - 2 = 0$

$2 a + 5 = 0$

Set $3 a - 2$ equal to $0$ and solve for $a$ .

Set $3 a - 2$ equal to $0$ .

$3 a - 2 = 0$

Solve $3 a - 2 = 0$ for $a$ .

Add $2$ to both sides of the equation.

$3 a = 2$

Divide each term in $3 a = 2$ by $3$ and simplify.

Divide each term in $3 a = 2$ by $3$ .

$\frac{3 a}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 a}{3} = \frac{2}{3}$

Divide $a$ by $1$ .

$a = \frac{2}{3}$

Set $2 a + 5$ equal to $0$ and solve for $a$ .

Set $2 a + 5$ equal to $0$ .

$2 a + 5 = 0$

Solve $2 a + 5 = 0$ for $a$ .

Subtract $5$ from both sides of the equation.

$2 a = - 5$

Divide each term in $2 a = - 5$ by $2$ and simplify.

Divide each term in $2 a = - 5$ by $2$ .

$\frac{2 a}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 5}{2}$

Divide $a$ by $1$ .

$a = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$a = - \frac{5}{2}$

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = 0.\overline{6}, - 2.5$

Mixed Number Form:

$a = \frac{2}{3}, - 2 \frac{1}{2}$

Solve for a 18a^2-30=-33a

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