Find All Complex Solutions 3x^3-2x^2-15x+10=0

$3 x^{3} - 2 x^{2} - 15 x + 10 = 0$

Step 1

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{3} - 2 x^{2}) - 15 x + 10 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

Step 2

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

$3 x - 2 = 0$

$x^{2} - 5 = 0$

Step 3

Set $3 x - 2$ equal to $0$ and solve for $x$ .

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Solve $3 x - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$3 x = 2$

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 4

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Solve $x^{2} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$x^{2} = 5$

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

$x = \pm \sqrt{5}$

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.

$x = \sqrt{5}$

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

$x = - \sqrt{5}$

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

$x = \sqrt{5}, - \sqrt{5}$

Step 5

The final solution is all the values that make $(3 x - 2) (x^{2} - 5) = 0$ true.

$x = \frac{2}{3}, \sqrt{5}, - \sqrt{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{3}, \sqrt{5}, - \sqrt{5}$

Decimal Form:

$x = 0.\overline{6}, 2.23606797 \dots, - 2.23606797 \dots$

Find All Complex Solutions 3x^3-2x^2-15x+10=0

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