Solve by Factoring x^4+2x^3=10x^2

$x^{4} + 2 x^{3} = 10 x^{2}$

Step 1

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

$x^{4} + 2 x^{3} - 10 x^{2} = 0$

Step 2

Factor $x^{2}$ out of $x^{4} + 2 x^{3} - 10 x^{2}$ .

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

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

Factor $x^{2}$ out of $2 x^{3}$ .

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

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

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

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

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

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

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

Step 3

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

$x^{2} = 0$

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

Step 4

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

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

$x^{2} = 0$

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

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

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

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

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

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 2$ , and $c = - 10$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 1 \cdot - 10$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 10$ .

$x = \frac{- 2 \pm \sqrt{4 + 40}}{2 \cdot 1}$

Add $4$ and $40$ .

$x = \frac{- 2 \pm \sqrt{44}}{2 \cdot 1}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{- 2 \pm \sqrt{4 (11)}}{2 \cdot 1}$

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{11}}{2}$

Simplify $\frac{- 2 \pm 2 \sqrt{11}}{2}$ .

$x = - 1 \pm \sqrt{11}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

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

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

Multiply $- 4 \cdot 1 \cdot - 10$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 10$ .

$x = \frac{- 2 \pm \sqrt{4 + 40}}{2 \cdot 1}$

Add $4$ and $40$ .

$x = \frac{- 2 \pm \sqrt{44}}{2 \cdot 1}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{- 2 \pm \sqrt{4 (11)}}{2 \cdot 1}$

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{11}}{2}$

Simplify $\frac{- 2 \pm 2 \sqrt{11}}{2}$ .

$x = - 1 \pm \sqrt{11}$

Change the $\pm$ to $+$ .

$x = - 1 + \sqrt{11}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

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

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

Multiply $- 4 \cdot 1 \cdot - 10$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 10$ .

$x = \frac{- 2 \pm \sqrt{4 + 40}}{2 \cdot 1}$

Add $4$ and $40$ .

$x = \frac{- 2 \pm \sqrt{44}}{2 \cdot 1}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{- 2 \pm \sqrt{4 (11)}}{2 \cdot 1}$

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{11}}{2}$

Simplify $\frac{- 2 \pm 2 \sqrt{11}}{2}$ .

$x = - 1 \pm \sqrt{11}$

Change the $\pm$ to $-$ .

$x = - 1 - \sqrt{11}$

The final answer is the combination of both solutions.

$x = - 1 + \sqrt{11}, - 1 - \sqrt{11}$

Step 6

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

$x = 0, - 1 + \sqrt{11}, - 1 - \sqrt{11}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 0, - 1 + \sqrt{11}, - 1 - \sqrt{11}$

Decimal Form:

$x = 0, 2.31662479 \dots, - 4.31662479 \dots$

Solve by Factoring x^4+2x^3=10x^2

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