Solve for r 9r^2+21r+9=-r^2

$9 r^{2} + 21 r + 9 = - r^{2}$

Step 1

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

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

$9 r^{2} + 21 r + 9 + r^{2} = 0$

Add $9 r^{2}$ and $r^{2}$ .

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

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

Factor $21$ out of $21 r$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 3

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

$5 r + 3 = 0$

$2 r + 3 = 0$

Step 4

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

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

$5 r + 3 = 0$

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

Subtract $3$ from both sides of the equation.

$5 r = - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $r$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 5

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

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

$2 r + 3 = 0$

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

Subtract $3$ from both sides of the equation.

$2 r = - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $r$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$r = - 0.6, - 1.5$

Mixed Number Form:

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

Solve for r 9r^2+21r+9=-r^2

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