Solve for q 3q^2-5q-6=-3q^2

$3 q^{2} - 5 q - 6 = - 3 q^{2}$

Step 1

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

Add $3 q^{2}$ to both sides of the equation.

$3 q^{2} - 5 q - 6 + 3 q^{2} = 0$

Add $3 q^{2}$ and $3 q^{2}$ .

$6 q^{2} - 5 q - 6 = 0$

Step 2

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 $a \cdot c = 6 \cdot - 6 = - 36$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 q$ .

$6 q^{2} - 5 q - 6 = 0$

Rewrite $- 5$ as $4$ plus $- 9$

$6 q^{2} + (4 - 9) q - 6 = 0$

Apply the distributive property.

$6 q^{2} + 4 q - 9 q - 6 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 q^{2} + 4 q) - 9 q - 6 = 0$

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

$2 q (3 q + 2) - 3 (3 q + 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 q + 2$ .

$(3 q + 2) (2 q - 3) = 0$

Step 3

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

$3 q + 2 = 0$

$2 q - 3 = 0$

Step 4

Set $3 q + 2$ equal to $0$ and solve for $q$ .

Set $3 q + 2$ equal to $0$ .

$3 q + 2 = 0$

Solve $3 q + 2 = 0$ for $q$ .

Subtract $2$ from both sides of the equation.

$3 q = - 2$

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

Divide each term in $3 q = - 2$ by $3$ .

$\frac{3 q}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 q}{3} = \frac{- 2}{3}$

Divide $q$ by $1$ .

$q = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$q = - \frac{2}{3}$

Step 5

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

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

$2 q - 3 = 0$

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

Add $3$ to both sides of the equation.

$2 q = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $q$ by $1$ .

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

Step 6

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

$q = - \frac{2}{3}, \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$q = - \frac{2}{3}, \frac{3}{2}$

Decimal Form:

$q = - 0.\overline{6}, 1.5$

Mixed Number Form:

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

Solve for q 3q^2-5q-6=-3q^2

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