Algebra Examples

$3 \sec^{2} (x) - 2 \tan^{2} (x) - 4 = 0$

Step 1

Replace the $3 \sec^{2} (x)$ with $3 (1 + \tan^{2} (x))$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$3 (1 + \tan^{2} (x)) - 2 \tan^{2} (x) - 4 = 0$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$3 \cdot 1 + 3 \tan^{2} (x) - 2 \tan^{2} (x) - 4 = 0$

Step 2.2

Multiply $3$ by $1$ .

$3 + 3 \tan^{2} (x) - 2 \tan^{2} (x) - 4 = 0$

Step 3

Simplify by adding terms.

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Step 3.1

Subtract $4$ from $3$ .

$3 \tan^{2} (x) - 2 \tan^{2} (x) - 1 = 0$

Step 3.2

Subtract $2 \tan^{2} (x)$ from $3 \tan^{2} (x)$ .

$\tan^{2} (x) - 1 = 0$

Step 4

Add $1$ to both sides of the equation.

$\tan^{2} (x) = 1$

Step 5

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

$\tan (x) = \pm \sqrt{1}$

Step 6

Any root of $1$ is $1$ .

$\tan (x) = \pm 1$

Step 7

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

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Step 7.1

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

$\tan (x) = 1$

Step 7.2

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

$\tan (x) = - 1$

Step 7.3

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

$\tan (x) = 1, - 1$

Step 8

Set up each of the solutions to solve for $x$ .

$\tan (x) = 1$

$\tan (x) = - 1$

Step 9

Solve for $x$ in $\tan (x) = 1$ .

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Step 9.1

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (1)$

Step 9.2

Simplify the right side.

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Step 9.2.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 9.3

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{4}$

Step 9.4

Simplify $π + \frac{π}{4}$ .

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Step 9.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 9.4.2

Combine fractions.

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Step 9.4.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 9.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 9.4.3

Simplify the numerator.

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Step 9.4.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 9.4.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 9.5

Find the period of $\tan (x)$ .

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Step 9.5.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 9.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 9.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 9.5.4

Divide $π$ by $1$ .

$π$

Step 9.6

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 10

Solve for $x$ in $\tan (x) = - 1$ .

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Step 10.1

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (- 1)$

Step 10.2

Simplify the right side.

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Step 10.2.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 10.3

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - \frac{π}{4} - π$

Step 10.4

Simplify the expression to find the second solution.

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Step 10.4.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 10.4.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 10.5

Find the period of $\tan (x)$ .

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Step 10.5.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 10.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 10.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 10.5.4

Divide $π$ by $1$ .

$π$

Step 10.6

Add $π$ to every negative angle to get positive angles.

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Step 10.6.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 10.6.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 10.6.3

Combine fractions.

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Step 10.6.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 10.6.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 10.6.4

Simplify the numerator.

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Step 10.6.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 10.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 10.6.5

List the new angles.

$x = \frac{3 π}{4}$

Step 10.7

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 11

List all of the solutions.

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n, \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 12

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$