Trigonometry Examples

$t = \sqrt{\frac{8 t + 24}{16}}$

Step 1

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

$\sqrt{\frac{8 t + 24}{16}} = t$

Step 2

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

${\sqrt{\frac{8 t + 24}{16}}}^{2} = t^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{8 t + 24}{16}}$ as ${(\frac{8 t + 24}{16})}^{\frac{1}{2}}$ .

${({(\frac{8 t + 24}{16})}^{\frac{1}{2}})}^{2} = t^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(\frac{8 t + 24}{16})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(\frac{8 t + 24}{16})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(\frac{8 t + 24}{16})}^{\frac{1}{2} \cdot 2} = t^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(\frac{8 t + 24}{16})}^{\frac{1}{2} \cdot 2} = t^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{8 t + 24}{16})}^{1} = t^{2}$

Step 3.2.1.2

Cancel the common factor of $8 t + 24$ and $16$ .

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Step 3.2.1.2.1

Factor $8$ out of $8 t$ .

${(\frac{8 (t) + 24}{16})}^{1} = t^{2}$

Step 3.2.1.2.2

Factor $8$ out of $24$ .

${(\frac{8 t + 8 \cdot 3}{16})}^{1} = t^{2}$

Step 3.2.1.2.3

Factor $8$ out of $8 t + 8 \cdot 3$ .

${(\frac{8 (t + 3)}{16})}^{1} = t^{2}$

Step 3.2.1.2.4

Cancel the common factors.

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Step 3.2.1.2.4.1

Factor $8$ out of $16$ .

${(\frac{8 (t + 3)}{8 (2)})}^{1} = t^{2}$

Step 3.2.1.2.4.2

Cancel the common factor.

${(\frac{8 (t + 3)}{8 \cdot 2})}^{1} = t^{2}$

Step 3.2.1.2.4.3

Rewrite the expression.

${(\frac{t + 3}{2})}^{1} = t^{2}$

Step 3.2.1.3

Simplify.

$\frac{t + 3}{2} = t^{2}$

Step 4

Solve for $t$ .

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Step 4.1

Multiply both sides by $2$ .

$\frac{t + 3}{2} \cdot 2 = t^{2} \cdot 2$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{t + 3}{2} \cdot 2 = t^{2} \cdot 2$

Step 4.2.1.1.2

Rewrite the expression.

$t + 3 = t^{2} \cdot 2$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Move $2$ to the left of $t^{2}$ .

$t + 3 = 2 t^{2}$

Step 4.3

Solve for $t$ .

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Step 4.3.1

Subtract $2 t^{2}$ from both sides of the equation.

$t + 3 - 2 t^{2} = 0$

Step 4.3.2

Factor the left side of the equation.

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Step 4.3.2.1

Factor $- 1$ out of $t + 3 - 2 t^{2}$ .

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Step 4.3.2.1.1

Reorder the expression.

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Step 4.3.2.1.1.1

Move $3$ .

$t - 2 t^{2} + 3 = 0$

Step 4.3.2.1.1.2

Reorder $t$ and $- 2 t^{2}$ .

$- 2 t^{2} + t + 3 = 0$

Step 4.3.2.1.2

Factor $- 1$ out of $- 2 t^{2}$ .

$- (2 t^{2}) + t + 3 = 0$

Step 4.3.2.1.3

Factor $- 1$ out of $t$ .

$- (2 t^{2}) - 1 (- t) + 3 = 0$

Step 4.3.2.1.4

Rewrite $3$ as $- 1 (- 3)$ .

$- (2 t^{2}) - 1 (- t) - 1 \cdot - 3 = 0$

Step 4.3.2.1.5

Factor $- 1$ out of $- (2 t^{2}) - 1 (- t)$ .

$- (2 t^{2} - t) - 1 \cdot - 3 = 0$

Step 4.3.2.1.6

Factor $- 1$ out of $- (2 t^{2} - t) - 1 (- 3)$ .

$- (2 t^{2} - t - 3) = 0$

Step 4.3.2.2

Factor.

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Step 4.3.2.2.1

Factor by grouping.

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Step 4.3.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 1$ .

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Step 4.3.2.2.1.1.1

Factor $- 1$ out of $- t$ .

$- (2 t^{2} - (t) - 3) = 0$

Step 4.3.2.2.1.1.2

Rewrite $- 1$ as $2$ plus $- 3$

$- (2 t^{2} + (2 - 3) t - 3) = 0$

Step 4.3.2.2.1.1.3

Apply the distributive property.

$- (2 t^{2} + 2 t - 3 t - 3) = 0$

Step 4.3.2.2.1.2

Factor out the greatest common factor from each group.

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Step 4.3.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((2 t^{2} + 2 t) - 3 t - 3) = 0$

Step 4.3.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (2 t (t + 1) - 3 (t + 1)) = 0$

Step 4.3.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $t + 1$ .

$- ((t + 1) (2 t - 3)) = 0$

Step 4.3.2.2.2

Remove unnecessary parentheses.

$- (t + 1) (2 t - 3) = 0$

Step 4.3.3

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

$t + 1 = 0$

$2 t - 3 = 0$

Step 4.3.4

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

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Step 4.3.4.1

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

$t + 1 = 0$

Step 4.3.4.2

Subtract $1$ from both sides of the equation.

$t = - 1$

Step 4.3.5

Set $2 t - 3$ equal to $0$ and solve for $t$ .

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Step 4.3.5.1

Set $2 t - 3$ equal to $0$ .

$2 t - 3 = 0$

Step 4.3.5.2

Solve $2 t - 3 = 0$ for $t$ .

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Step 4.3.5.2.1

Add $3$ to both sides of the equation.

$2 t = 3$

Step 4.3.5.2.2

Divide each term in $2 t = 3$ by $2$ and simplify.

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Step 4.3.5.2.2.1

Divide each term in $2 t = 3$ by $2$ .

$\frac{2 t}{2} = \frac{3}{2}$

Step 4.3.5.2.2.2

Simplify the left side.

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Step 4.3.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{3}{2}$

Step 4.3.5.2.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{3}{2}$

Step 4.3.6

The final solution is all the values that make $- (t + 1) (2 t - 3) = 0$ true.

$t = - 1, \frac{3}{2}$

Step 5

Exclude the solutions that do not make $t = \sqrt{\frac{8 t + 24}{16}}$ true.

$t = \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{3}{2}$

Decimal Form:

$t = 1.5$

Mixed Number Form:

$t = 1 \frac{1}{2}$