Algebra Examples

$x^{4} = 5 x^{3} - 9 x^{2}$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$5 x^{3} - 9 x^{2} = x^{4}$

Step 2

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

$5 x^{3} - 9 x^{2} - x^{4} = 0$

Step 3

Factor $- x^{2}$ out of $5 x^{3} - 9 x^{2} - x^{4}$ .

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Step 3.1

Reorder the expression.

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Step 3.1.1

Move $- 9 x^{2}$ .

$5 x^{3} - x^{4} - 9 x^{2} = 0$

Step 3.1.2

Reorder $5 x^{3}$ and $- x^{4}$ .

$- x^{4} + 5 x^{3} - 9 x^{2} = 0$

Step 3.2

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

$- x^{2} x^{2} + 5 x^{3} - 9 x^{2} = 0$

Step 3.3

Factor $- x^{2}$ out of $5 x^{3}$ .

$- x^{2} x^{2} - x^{2} (- 5 x) - 9 x^{2} = 0$

Step 3.4

Factor $- x^{2}$ out of $- 9 x^{2}$ .

$- x^{2} x^{2} - x^{2} (- 5 x) - x^{2} \cdot 9 = 0$

Step 3.5

Factor $- x^{2}$ out of $- x^{2} (x^{2}) - x^{2} (- 5 x)$ .

$- x^{2} (x^{2} - 5 x) - x^{2} \cdot 9 = 0$

Step 3.6

Factor $- x^{2}$ out of $- x^{2} (x^{2} - 5 x) - x^{2} (9)$ .

$- x^{2} (x^{2} - 5 x + 9) = 0$

Step 4

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

$x^{2} = 0$

$x^{2} - 5 x + 9 = 0$

Step 5

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

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Step 5.1

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

$x^{2} = 0$

Step 5.2

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

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Step 5.2.1

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

$x = \pm \sqrt{0}$

Step 5.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.2.2.1

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

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

Step 5.2.2.2

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

$x = \pm 0$

Step 5.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

Set $x^{2} - 5 x + 9$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{2} - 5 x + 9$ equal to $0$ .

$x^{2} - 5 x + 9 = 0$

Step 6.2

Solve $x^{2} - 5 x + 9 = 0$ for $x$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

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

Step 6.2.2

Substitute the values $a = 1$ , $b = - 5$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 9}}{2 \cdot 1}$

Step 6.2.3.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{5 \pm \sqrt{25 - 36}}{2 \cdot 1}$

Step 6.2.3.1.3

Subtract $36$ from $25$ .

$x = \frac{5 \pm \sqrt{- 11}}{2 \cdot 1}$

Step 6.2.3.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Step 6.2.3.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Step 6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{11}}{2 \cdot 1}$

Step 6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm i \sqrt{11}}{2}$

Step 6.2.4

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{11}}{2}, \frac{5 - i \sqrt{11}}{2}$

Step 7

The final solution is all the values that make $- x^{2} (x^{2} - 5 x + 9) = 0$ true.

$x = 0, \frac{5 + i \sqrt{11}}{2}, \frac{5 - i \sqrt{11}}{2}$