Find the Roots (Zeros) x^4+x^3=4x^2+4x

$x^{4} + x^{3} = 4 x^{2} + 4 x$

Step 1

Move all terms containing $x$ to the left side of the equation.

Subtract $4 x^{2}$ from both sides of the equation.

$x^{4} + x^{3} - 4 x^{2} = 4 x$

Subtract $4 x$ from both sides of the equation.

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

Step 2

Factor the left side of the equation.

Factor $x$ out of $x^{4} + x^{3} - 4 x^{2} - 4 x$ .

Factor $x$ out of $x^{4}$ .

$x \cdot x^{3} + x^{3} - 4 x^{2} - 4 x = 0$

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

$x \cdot x^{3} + x \cdot x^{2} - 4 x^{2} - 4 x = 0$

Factor $x$ out of $- 4 x^{2}$ .

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

Factor $x$ out of $- 4 x$ .

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

Factor $x$ out of $x \cdot x^{3} + x \cdot x^{2}$ .

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

Factor $x$ out of $x (x^{3} + x^{2}) + x (- 4 x)$ .

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

Factor $x$ out of $x (x^{3} + x^{2} - 4 x) + x \cdot - 4$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

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

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 $b = 2$ .

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

Step 3

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

$x = 0$

$x + 1 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

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 6

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 7

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 8

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

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

Step 9

Find the Roots (Zeros) x^4+x^3=4x^2+4x

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