Solve for x x^3+13x=10x^2

$x^{3} + 13 x = 10 x^{2}$

Subtract $10 x^{2}$ from both sides of the equation.

$x^{3} + 13 x - 10 x^{2} = 0$

Factor the left side of the equation.

Factor $x$ out of $x^{3} + 13 x - 10 x^{2}$ .

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

$x \cdot x^{2} + 13 x - 10 x^{2} = 0$

Factor $x$ out of $13 x$ .

$x \cdot x^{2} + x \cdot 13 - 10 x^{2} = 0$

Factor $x$ out of $- 10 x^{2}$ .

$x \cdot x^{2} + x \cdot 13 + x (- 10 x) = 0$

Factor $x$ out of $x \cdot x^{2} + x \cdot 13$ .

$x (x^{2} + 13) + x (- 10 x) = 0$

Factor $x$ out of $x (x^{2} + 13) + x (- 10 x)$ .

$x (x^{2} + 13 - 10 x) = 0$

Reorder terms.

$x (x^{2} - 10 x + 13) = 0$

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

$x = 0$

$x^{2} - 10 x + 13 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x^{2} - 10 x + 13$ equal to $0$ and solve for $x$ .

Set $x^{2} - 10 x + 13$ equal to $0$ .

$x^{2} - 10 x + 13 = 0$

Solve $x^{2} - 10 x + 13 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 10$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 13}}{2 \cdot 1}$

Multiply $- 4$ by $13$ .

$x = \frac{10 \pm \sqrt{100 - 52}}{2 \cdot 1}$

Subtract $52$ from $100$ .

$x = \frac{10 \pm \sqrt{48}}{2 \cdot 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{10 \pm \sqrt{16 (3)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{10 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{10 \pm 4 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 4 \sqrt{3}}{2}$

Simplify $\frac{10 \pm 4 \sqrt{3}}{2}$ .

$x = 5 \pm 2 \sqrt{3}$

The final answer is the combination of both solutions.

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

The final solution is all the values that make $x (x^{2} - 10 x + 13) = 0$ true.

$x = 0, 5 + 2 \sqrt{3}, 5 - 2 \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$x = 0, 5 + 2 \sqrt{3}, 5 - 2 \sqrt{3}$

Decimal Form:

$x = 0, 8.46410161 \dots, 1.53589838 \dots$

Solve for x x^3+13x=10x^2

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