Solve for a 12a^2+28a=-8

$12 a^{2} + 28 a = - 8$

Step 1

Add $8$ to both sides of the equation.

$12 a^{2} + 28 a + 8 = 0$

Step 2

Factor the left side of the equation.

Factor $4$ out of $12 a^{2} + 28 a + 8$ .

Factor $4$ out of $12 a^{2}$ .

$4 (3 a^{2}) + 28 a + 8 = 0$

Factor $4$ out of $28 a$ .

$4 (3 a^{2}) + 4 (7 a) + 8 = 0$

Factor $4$ out of $8$ .

$4 (3 a^{2}) + 4 (7 a) + 4 (2) = 0$

Factor $4$ out of $4 (3 a^{2}) + 4 (7 a)$ .

$4 (3 a^{2} + 7 a) + 4 (2) = 0$

Factor $4$ out of $4 (3 a^{2} + 7 a) + 4 (2)$ .

$4 (3 a^{2} + 7 a + 2) = 0$

Factor.

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 $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = 7$ .

Factor $7$ out of $7 a$ .

$4 (3 a^{2} + 7 (a) + 2) = 0$

Rewrite $7$ as $1$ plus $6$

$4 (3 a^{2} + (1 + 6) a + 2) = 0$

Apply the distributive property.

$4 (3 a^{2} + 1 a + 6 a + 2) = 0$

Multiply $a$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$4 (a (3 a + 1) + 2 (3 a + 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 a + 1$ .

$4 ((3 a + 1) (a + 2)) = 0$

Remove unnecessary parentheses.

$4 (3 a + 1) (a + 2) = 0$

Step 3

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

$3 a + 1 = 0$

$a + 2 = 0$

Step 4

Set $3 a + 1$ equal to $0$ and solve for $a$ .

Set $3 a + 1$ equal to $0$ .

$3 a + 1 = 0$

Solve $3 a + 1 = 0$ for $a$ .

Subtract $1$ from both sides of the equation.

$3 a = - 1$

Divide each term in $3 a = - 1$ by $3$ and simplify.

Divide each term in $3 a = - 1$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 5

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

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

$a + 2 = 0$

Subtract $2$ from both sides of the equation.

$a = - 2$

Step 6

The final solution is all the values that make $4 (3 a + 1) (a + 2) = 0$ true.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = - 0.\overline{3}, - 2$

Solve for a 12a^2+28a=-8

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