Solve by Factoring x^3+3x^2=25x+75

$x^{3} + 3 x^{2} = 25 x + 75$

Step 1

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

Subtract $25 x$ from both sides of the equation.

$x^{3} + 3 x^{2} - 25 x = 75$

Subtract $75$ from both sides of the equation.

$x^{3} + 3 x^{2} - 25 x - 75 = 0$

Step 2

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{3} + 3 x^{2}) - 25 x - 75 = 0$

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

$x^{2} (x + 3) - 25 (x + 3) = 0$

Step 3

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

$(x + 3) (x^{2} - 25) = 0$

Step 4

Rewrite $25$ as $5^{2}$ .

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

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 $b = 5$ .

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

Remove unnecessary parentheses.

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

Step 6

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

$x + 3 = 0$

$x + 5 = 0$

$x - 5 = 0$

Step 7

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 8

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

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

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 9

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 10

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

$x = - 3, - 5, 5$

Solve by Factoring x^3+3x^2=25x+75

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