Precalculus Examples

$| x^{2} - 8 x | = 9 x$

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x^{2} - 8 x = \pm 9 x$

Step 2

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

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Step 2.1

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

$x^{2} - 8 x = 9 x$

Step 2.2

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

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Step 2.2.1

Subtract $9 x$ from both sides of the equation.

$x^{2} - 8 x - 9 x = 0$

Step 2.2.2

Subtract $9 x$ from $- 8 x$ .

$x^{2} - 17 x = 0$

Step 2.3

Factor $x$ out of $x^{2} - 17 x$ .

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Step 2.3.1

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

$x \cdot x - 17 x = 0$

Step 2.3.2

Factor $x$ out of $- 17 x$ .

$x \cdot x + x \cdot - 17 = 0$

Step 2.3.3

Factor $x$ out of $x \cdot x + x \cdot - 17$ .

$x (x - 17) = 0$

Step 2.4

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

$x = 0$

$x - 17 = 0$

Step 2.5

Set $x$ equal to $0$ .

$x = 0$

Step 2.6

Set $x - 17$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 17$ equal to $0$ .

$x - 17 = 0$

Step 2.6.2

Add $17$ to both sides of the equation.

$x = 17$

Step 2.7

The final solution is all the values that make $x (x - 17) = 0$ true.

$x = 0, 17$

Step 2.8

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

$x^{2} - 8 x = - 9 x$

Step 2.9

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

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Step 2.9.1

Add $9 x$ to both sides of the equation.

$x^{2} - 8 x + 9 x = 0$

Step 2.9.2

Add $- 8 x$ and $9 x$ .

$x^{2} + x = 0$

Step 2.10

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

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Step 2.10.1

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

$x \cdot x + x = 0$

Step 2.10.2

Raise $x$ to the power of $1$ .

$x \cdot x + x = 0$

Step 2.10.3

Factor $x$ out of $x^{1}$ .

$x \cdot x + x \cdot 1 = 0$

Step 2.10.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$x (x + 1) = 0$

Step 2.11

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

$x = 0$

$x + 1 = 0$

Step 2.12

Set $x$ equal to $0$ .

$x = 0$

Step 2.13

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.13.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.13.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.14

The final solution is all the values that make $x (x + 1) = 0$ true.

$x = 0, - 1$

Step 2.15

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

$x = 0, 17, - 1$

Step 3

Exclude the solutions that do not make $| x^{2} - 8 x | = 9 x$ true.

$x = 0, 17$