Trigonometry Examples

$f (x) = \sqrt{3 x^{2}} - 8 + 6$ , $f (x) = 8$

Step 1

Substitute $8$ for $f (x)$ .

$8 = \sqrt{3 x^{2}} - 8 + 6$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $\sqrt{3 x^{2}} - 8 + 6 = 8$ .

$\sqrt{3 x^{2}} - 8 + 6 = 8$

Step 2.2

Solve for $\sqrt{3 x^{2}}$ .

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Step 2.2.1

Add $- 8$ and $6$ .

$\sqrt{3 x^{2}} - 2 = 8$

Step 2.2.2

Move all terms not containing $\sqrt{3 x^{2}}$ to the right side of the equation.

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Step 2.2.2.1

Add $2$ to both sides of the equation.

$\sqrt{3 x^{2}} = 8 + 2$

Step 2.2.2.2

Add $8$ and $2$ .

$\sqrt{3 x^{2}} = 10$

Step 2.3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{3 x^{2}}}^{2} = 10^{2}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x^{2}}$ as ${(3 x^{2})}^{\frac{1}{2}}$ .

${({(3 x^{2})}^{\frac{1}{2}})}^{2} = 10^{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Simplify ${({(3 x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.4.2.1.1

Multiply the exponents in ${({(3 x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(3 x^{2})}^{\frac{1}{2} \cdot 2} = 10^{2}$

Step 2.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

${(3 x^{2})}^{\frac{1}{2} \cdot 2} = 10^{2}$

Step 2.4.2.1.1.2.2

Rewrite the expression.

${(3 x^{2})}^{1} = 10^{2}$

Step 2.4.2.1.2

Simplify.

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Raise $10$ to the power of $2$ .

$3 x^{2} = 100$

Step 2.5

Solve for $x$ .

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Step 2.5.1

Divide each term in $3 x^{2} = 100$ by $3$ and simplify.

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Step 2.5.1.1

Divide each term in $3 x^{2} = 100$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{100}{3}$

Step 2.5.1.2

Simplify the left side.

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Step 2.5.1.2.1

Cancel the common factor of $3$ .

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Step 2.5.1.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{100}{3}$

Step 2.5.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{100}{3}$

Step 2.5.2

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

$x = \pm \sqrt{\frac{100}{3}}$

Step 2.5.3

Simplify $\pm \sqrt{\frac{100}{3}}$ .

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Step 2.5.3.1

Rewrite $\sqrt{\frac{100}{3}}$ as $\frac{\sqrt{100}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{100}}{\sqrt{3}}$

Step 2.5.3.2

Simplify the numerator.

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Step 2.5.3.2.1

Rewrite $100$ as $10^{2}$ .

$x = \pm \frac{\sqrt{10^{2}}}{\sqrt{3}}$

Step 2.5.3.2.2

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

$x = \pm \frac{10}{\sqrt{3}}$

Step 2.5.3.3

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.5.3.4

Combine and simplify the denominator.

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Step 2.5.3.4.1

Multiply $\frac{10}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{10 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.5.3.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{10 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.5.3.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{10 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.5.3.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{10 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.5.3.4.5

Add $1$ and $1$ .

$x = \pm \frac{10 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.5.3.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.5.3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{10 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.5.3.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{10 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.5.3.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.5.3.4.6.4

Cancel the common factor of $2$ .

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Step 2.5.3.4.6.4.1

Cancel the common factor.

$x = \pm \frac{10 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.5.3.4.6.4.2

Rewrite the expression.

$x = \pm \frac{10 \sqrt{3}}{3^{1}}$

Step 2.5.3.4.6.5

Evaluate the exponent.

$x = \pm \frac{10 \sqrt{3}}{3}$

Step 2.5.4

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

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Step 2.5.4.1

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

$x = \frac{10 \sqrt{3}}{3}$

Step 2.5.4.2

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

$x = - \frac{10 \sqrt{3}}{3}$

Step 2.5.4.3

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

$x = \frac{10 \sqrt{3}}{3}, - \frac{10 \sqrt{3}}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{10 \sqrt{3}}{3}, - \frac{10 \sqrt{3}}{3}$

Decimal Form:

$x = 5.77350269 \dots, - 5.77350269 \dots$