Solve by Factoring -12q=4q^2+8

$- 12 q = 4 q^{2} + 8$

Step 1

Move all the expressions to the left side of the equation.

Subtract $4 q^{2}$ from both sides of the equation.

$- 12 q - 4 q^{2} = 8$

Subtract $8$ from both sides of the equation.

$- 12 q - 4 q^{2} - 8 = 0$

Step 2

Factor $4$ out of $- 12 q - 4 q^{2} - 8$ .

Factor $4$ out of $- 12 q$ .

$4 (- 3 q) - 4 q^{2} - 8 = 0$

Factor $4$ out of $- 4 q^{2}$ .

$4 (- 3 q) + 4 (- q^{2}) - 8 = 0$

Factor $4$ out of $- 8$ .

$4 (- 3 q) + 4 (- q^{2}) + 4 (- 2) = 0$

Factor $4$ out of $4 (- 3 q) + 4 (- q^{2})$ .

$4 (- 3 q - q^{2}) + 4 (- 2) = 0$

Factor $4$ out of $4 (- 3 q - q^{2}) + 4 (- 2)$ .

$4 (- 3 q - q^{2} - 2) = 0$

Step 3

Let $u = q$ . Substitute $u$ for all occurrences of $q$ .

$4 (- 3 u - u^{2} - 2) = 0$

Step 4

Factor by grouping.

Reorder terms.

$4 (- u^{2} - 3 u - 2) = 0$

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 $a \cdot c = - 1 \cdot - 2 = 2$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 u$ .

$4 (- u^{2} - 3 u - 2) = 0$

Rewrite $- 3$ as $- 1$ plus $- 2$

$4 (- u^{2} + (- 1 - 2) u - 2) = 0$

Apply the distributive property.

$4 (- u^{2} - 1 u - 2 u - 2) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((- u^{2} - 1 u) - 2 u - 2) = 0$

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

$4 (u (- u - 1) + 2 (- u - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $- u - 1$ .

$4 ((- u - 1) (u + 2)) = 0$

Step 5

Factor.

Replace all occurrences of $u$ with $q$ .

$4 ((- q - 1) (q + 2)) = 0$

Remove unnecessary parentheses.

$4 (- q - 1) (q + 2) = 0$

Step 6

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

$- q - 1 = 0$

$q + 2 = 0$

Step 7

Set $- q - 1$ equal to $0$ and solve for $q$ .

Set $- q - 1$ equal to $0$ .

$- q - 1 = 0$

Solve $- q - 1 = 0$ for $q$ .

Add $1$ to both sides of the equation.

$- q = 1$

Divide each term in $- q = 1$ by $- 1$ and simplify.

Divide each term in $- q = 1$ by $- 1$ .

$\frac{- q}{- 1} = \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{q}{1} = \frac{1}{- 1}$

Divide $q$ by $1$ .

$q = \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$q = - 1$

Step 8

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

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

$q + 2 = 0$

Subtract $2$ from both sides of the equation.

$q = - 2$

Step 9

The final solution is all the values that make $4 (- q - 1) (q + 2) = 0$ true.

$q = - 1, - 2$

Solve by Factoring -12q=4q^2+8

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