Solve for a 16a^2+21a+9=6a^2

$16 a^{2} + 21 a + 9 = 6 a^{2}$

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

Subtract $6 a^{2}$ from both sides of the equation.

$16 a^{2} + 21 a + 9 - 6 a^{2} = 0$

Subtract $6 a^{2}$ from $16 a^{2}$ .

$10 a^{2} + 21 a + 9 = 0$

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 $a \cdot c = 10 \cdot 9 = 90$ and whose sum is $b = 21$ .

Factor $21$ out of $21 a$ .

$10 a^{2} + 21 (a) + 9 = 0$

Rewrite $21$ as $6$ plus $15$

$10 a^{2} + (6 + 15) a + 9 = 0$

Apply the distributive property.

$10 a^{2} + 6 a + 15 a + 9 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(10 a^{2} + 6 a) + 15 a + 9 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $5 a + 3$ .

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

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

$5 a + 3 = 0$

$2 a + 3 = 0$

Set $5 a + 3$ equal to $0$ and solve for $a$ .

Set $5 a + 3$ equal to $0$ .

$5 a + 3 = 0$

Solve $5 a + 3 = 0$ for $a$ .

Subtract $3$ from both sides of the equation.

$5 a = - 3$

Divide each term in $5 a = - 3$ by $5$ and simplify.

Divide each term in $5 a = - 3$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

$2 a + 3 = 0$

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

Subtract $3$ from both sides of the equation.

$2 a = - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = - 0.6, - 1.5$

Mixed Number Form:

$a = - \frac{3}{5}, - 1 \frac{1}{2}$

Solve for a 16a^2+21a+9=6a^2

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