Trigonometry Examples

$| x^{2} - 2 x | = 3 x - 6$

Step 1

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

$x^{2} - 2 x = \pm (3 x - 6)$

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.

$x^{2} - 2 x = 3 x - 6$

Step 2.2

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

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Step 2.2.1

Subtract $3 x$ from both sides of the equation.

$x^{2} - 2 x - 3 x = - 6$

Step 2.2.2

Subtract $3 x$ from $- 2 x$ .

$x^{2} - 5 x = - 6$

Step 2.3

Add $6$ to both sides of the equation.

$x^{2} - 5 x + 6 = 0$

Step 2.4

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 2.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 2.4.2

Write the factored form using these integers.

$(x - 3) (x - 2) = 0$

Step 2.5

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

$x - 3 = 0$

$x - 2 = 0$

Step 2.6

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

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Step 2.6.1

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

$x - 3 = 0$

Step 2.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.7

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

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Step 2.7.1

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

$x - 2 = 0$

Step 2.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.8

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

$x = 3, 2$

Step 2.9

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

$x^{2} - 2 x = - (3 x - 6)$

Step 2.10

Simplify $- (3 x - 6)$ .

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Step 2.10.1

Apply the distributive property.

$x^{2} - 2 x = - (3 x) - - 6$

Step 2.10.2

Multiply.

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Step 2.10.2.1

Multiply $3$ by $- 1$ .

$x^{2} - 2 x = - 3 x - - 6$

Step 2.10.2.2

Multiply $- 1$ by $- 6$ .

$x^{2} - 2 x = - 3 x + 6$

Step 2.11

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

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Step 2.11.1

Add $3 x$ to both sides of the equation.

$x^{2} - 2 x + 3 x = 6$

Step 2.11.2

Add $- 2 x$ and $3 x$ .

$x^{2} + x = 6$

Step 2.12

Subtract $6$ from both sides of the equation.

$x^{2} + x - 6 = 0$

Step 2.13

Factor $x^{2} + x - 6$ using the AC method.

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Step 2.13.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 2.13.2

Write the factored form using these integers.

$(x - 2) (x + 3) = 0$

Step 2.14

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

$x - 2 = 0$

$x + 3 = 0$

Step 2.15

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

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Step 2.15.1

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

$x - 2 = 0$

Step 2.15.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.16

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.16.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.16.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.17

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

$x = 2, - 3$

Step 2.18

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

$x = 3, 2, - 3$

Step 3

Exclude the solutions that do not make $| x^{2} - 2 x | = 3 x - 6$ true.

$x = 3, 2$