Solve for a -5=9a^2+46a

$- 5 = 9 a^{2} + 46 a$

Step 1

Rewrite the equation as $9 a^{2} + 46 a = - 5$ .

$9 a^{2} + 46 a = - 5$

Step 2

Add $5$ to both sides of the equation.

$9 a^{2} + 46 a + 5 = 0$

Step 3

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 $a \cdot c = 9 \cdot 5 = 45$ and whose sum is $b = 46$ .

Factor $46$ out of $46 a$ .

$9 a^{2} + 46 (a) + 5 = 0$

Rewrite $46$ as $1$ plus $45$

$9 a^{2} + (1 + 45) a + 5 = 0$

Apply the distributive property.

$9 a^{2} + 1 a + 45 a + 5 = 0$

Multiply $a$ by $1$ .

$9 a^{2} + a + 45 a + 5 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 a^{2} + a) + 45 a + 5 = 0$

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

$a (9 a + 1) + 5 (9 a + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $9 a + 1$ .

$(9 a + 1) (a + 5) = 0$

Step 4

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

$9 a + 1 = 0$

$a + 5 = 0$

Step 5

Set $9 a + 1$ equal to $0$ and solve for $a$ .

Set $9 a + 1$ equal to $0$ .

$9 a + 1 = 0$

Solve $9 a + 1 = 0$ for $a$ .

Subtract $1$ from both sides of the equation.

$9 a = - 1$

Divide each term in $9 a = - 1$ by $9$ and simplify.

Divide each term in $9 a = - 1$ by $9$ .

$\frac{9 a}{9} = \frac{- 1}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 a}{9} = \frac{- 1}{9}$

Divide $a$ by $1$ .

$a = \frac{- 1}{9}$

Simplify the right side.

Move the negative in front of the fraction.

$a = - \frac{1}{9}$

Step 6

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

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

$a + 5 = 0$

Subtract $5$ from both sides of the equation.

$a = - 5$

Step 7

The final solution is all the values that make $(9 a + 1) (a + 5) = 0$ true.

$a = - \frac{1}{9}, - 5$

Step 8

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{1}{9}, - 5$

Decimal Form:

$a = - 0.\overline{1}, - 5$

Solve for a -5=9a^2+46a

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