Find the Roots (Zeros) 2c^4-6c^3=12c^2-36c

$2 c^{4} - 6 c^{3} = 12 c^{2} - 36 c$

Step 1

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

Subtract $12 c^{2}$ from both sides of the equation.

$2 c^{4} - 6 c^{3} - 12 c^{2} = - 36 c$

Add $36 c$ to both sides of the equation.

$2 c^{4} - 6 c^{3} - 12 c^{2} + 36 c = 0$

Step 2

Factor the left side of the equation.

Factor $2 c$ out of $2 c^{4} - 6 c^{3} - 12 c^{2} + 36 c$ .

Factor $2 c$ out of $2 c^{4}$ .

$2 c (c^{3}) - 6 c^{3} - 12 c^{2} + 36 c = 0$

Factor $2 c$ out of $- 6 c^{3}$ .

$2 c (c^{3}) + 2 c (- 3 c^{2}) - 12 c^{2} + 36 c = 0$

Factor $2 c$ out of $- 12 c^{2}$ .

$2 c (c^{3}) + 2 c (- 3 c^{2}) + 2 c (- 6 c) + 36 c = 0$

Factor $2 c$ out of $36 c$ .

$2 c (c^{3}) + 2 c (- 3 c^{2}) + 2 c (- 6 c) + 2 c (18) = 0$

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

$2 c (c^{3} - 3 c^{2}) + 2 c (- 6 c) + 2 c (18) = 0$

Factor $2 c$ out of $2 c (c^{3} - 3 c^{2}) + 2 c (- 6 c)$ .

$2 c (c^{3} - 3 c^{2} - 6 c) + 2 c (18) = 0$

Factor $2 c$ out of $2 c (c^{3} - 3 c^{2} - 6 c) + 2 c (18)$ .

$2 c (c^{3} - 3 c^{2} - 6 c + 18) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 c ((c^{3} - 3 c^{2}) - 6 c + 18) = 0$

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

$2 c (c^{2} (c - 3) - 6 (c - 3)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $c - 3$ .

$2 c ((c - 3) (c^{2} - 6)) = 0$

Remove unnecessary parentheses.

$2 c (c - 3) (c^{2} - 6) = 0$

Step 3

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

$c = 0$

$c - 3 = 0$

$c^{2} - 6 = 0$

Step 4

Set $c$ equal to $0$ .

$c = 0$

Step 5

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

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

$c - 3 = 0$

Add $3$ to both sides of the equation.

$c = 3$

Step 6

Set $c^{2} - 6$ equal to $0$ and solve for $c$ .

Set $c^{2} - 6$ equal to $0$ .

$c^{2} - 6 = 0$

Solve $c^{2} - 6 = 0$ for $c$ .

Add $6$ to both sides of the equation.

$c^{2} = 6$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$c = \pm \sqrt{6}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$c = \sqrt{6}$

Next, use the negative value of the $\pm$ to find the second solution.

$c = - \sqrt{6}$

The complete solution is the result of both the positive and negative portions of the solution.

$c = \sqrt{6}, - \sqrt{6}$

Step 7

The final solution is all the values that make $2 c (c - 3) (c^{2} - 6) = 0$ true.

$c = 0, 3, \sqrt{6}, - \sqrt{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$c = 0, 3, \sqrt{6}, - \sqrt{6}$

Decimal Form:

$c = 0, 3, 2.44948974 \dots, - 2.44948974 \dots$

Step 9

Find the Roots (Zeros) 2c^4-6c^3=12c^2-36c

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