Solve for x x^5-5x^3+4x=0

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

Step 1

Factor the left side of the equation.

Factor $x$ out of $x^{5} - 5 x^{3} + 4 x$ .

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 5 x^{3} + 4 x = 0$

Factor $x$ out of $- 5 x^{3}$ .

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

Factor $x$ out of $4 x$ .

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

Factor $x$ out of $x \cdot x^{4} + x (- 5 x^{2})$ .

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

Factor $x$ out of $x (x^{4} - 5 x^{2}) + x \cdot 4$ .

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

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

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

Factor $u^{2} - 5 u + 4$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $- 5$ .

$- 4, - 1$

Write the factored form using these integers.

$x ((u - 4) (u - 1)) = 0$

Replace all occurrences of $u$ with $x^{2}$ .

$x ((x^{2} - 4) (x^{2} - 1)) = 0$

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

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

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

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 $b = 1$ .

$x ((x + 2) (x - 2) ((x + 1) (x - 1))) = 0$

Remove unnecessary parentheses.

$x ((x + 2) (x - 2) (x + 1) (x - 1)) = 0$

Remove unnecessary parentheses.

$x (x + 2) (x - 2) (x + 1) (x - 1) = 0$

Step 2

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

$x = 0$

$x + 2 = 0$

$x - 2 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 6

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 8

The final solution is all the values that make $x (x + 2) (x - 2) (x + 1) (x - 1) = 0$ true.

$x = 0, - 2, 2, - 1, 1$

Solve for x x^5-5x^3+4x=0

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