Solve by Factoring 5x^3+4x^2=45x+36

$5 x^{3} + 4 x^{2} = 45 x + 36$

Step 1

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

Subtract $45 x$ from both sides of the equation.

$5 x^{3} + 4 x^{2} - 45 x = 36$

Subtract $36$ from both sides of the equation.

$5 x^{3} + 4 x^{2} - 45 x - 36 = 0$

Step 2

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 x^{3} + 4 x^{2}) - 45 x - 36 = 0$

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

$x^{2} (5 x + 4) - 9 (5 x + 4) = 0$

Step 3

Factor the polynomial by factoring out the greatest common factor, $5 x + 4$ .

$(5 x + 4) (x^{2} - 9) = 0$

Step 4

Rewrite $9$ as $3^{2}$ .

$(5 x + 4) (x^{2} - 3^{2}) = 0$

Step 5

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$(5 x + 4) ((x + 3) (x - 3)) = 0$

Remove unnecessary parentheses.

$(5 x + 4) (x + 3) (x - 3) = 0$

Step 6

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

$5 x + 4 = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 7

Set $5 x + 4$ equal to $0$ and solve for $x$ .

Set $5 x + 4$ equal to $0$ .

$5 x + 4 = 0$

Solve $5 x + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$5 x = - 4$

Divide each term in $5 x = - 4$ by $5$ and simplify.

Divide each term in $5 x = - 4$ by $5$ .

$\frac{5 x}{5} = \frac{- 4}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 4}{5}$

Divide $x$ by $1$ .

$x = \frac{- 4}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{4}{5}$

Step 8

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 9

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 10

The final solution is all the values that make $(5 x + 4) (x + 3) (x - 3) = 0$ true.

$x = - \frac{4}{5}, - 3, 3$

Solve by Factoring 5x^3+4x^2=45x+36

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