Solve by Factoring 25x^3-x=0

$25 x^{3} - x = 0$

Step 1

Factor $x$ out of $25 x^{3} - x$ .

Factor $x$ out of $25 x^{3}$ .

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

Factor $x$ out of $- x$ .

$x (25 x^{2}) + x \cdot - 1 = 0$

Factor $x$ out of $x (25 x^{2}) + x \cdot - 1$ .

$x (25 x^{2} - 1) = 0$

Step 2

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$x ({(5 x)}^{2} - 1) = 0$

Step 3

Rewrite $1$ as $1^{2}$ .

$x ({(5 x)}^{2} - 1^{2}) = 0$

Step 4

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 x$ and $b = 1$ .

$x ((5 x + 1) (5 x - 1)) = 0$

Remove unnecessary parentheses.

$x (5 x + 1) (5 x - 1) = 0$

Step 5

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

$x = 0$

$5 x + 1 = 0$

$5 x - 1 = 0$

Step 6

Set $x$ equal to $0$ .

$x = 0$

Step 7

Set $5 x + 1$ equal to $0$ and solve for $x$ .

Set $5 x + 1$ equal to $0$ .

$5 x + 1 = 0$

Solve $5 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$5 x = - 1$

Divide each term in $5 x = - 1$ by $5$ and simplify.

Divide each term in $5 x = - 1$ by $5$ .

$\frac{5 x}{5} = \frac{- 1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 1}{5}$

Divide $x$ by $1$ .

$x = \frac{- 1}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

Step 8

Set $5 x - 1$ equal to $0$ and solve for $x$ .

Set $5 x - 1$ equal to $0$ .

$5 x - 1 = 0$

Solve $5 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$5 x = 1$

Divide each term in $5 x = 1$ by $5$ and simplify.

Divide each term in $5 x = 1$ by $5$ .

$\frac{5 x}{5} = \frac{1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{1}{5}$

Divide $x$ by $1$ .

$x = \frac{1}{5}$

Step 9

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

$x = 0, - \frac{1}{5}, \frac{1}{5}$

Solve by Factoring 25x^3-x=0

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