Trigonometry Examples

y = x - 16 {(\frac{x}{\sqrt{2} \cdot 105})}^{2}

$y = x - 16 {(\frac{x}{\sqrt{2} \cdot 105})}^{2}$

Step 1

Rewrite the equation as

x - 16 {(\frac{x}{\sqrt{2} \cdot 105})}^{2} = y

$x - 16 {(\frac{x}{\sqrt{2} \cdot 105})}^{2} = y$ .

x - 16 {(\frac{x}{\sqrt{2} \cdot 105})}^{2} = y

$x - 16 {(\frac{x}{\sqrt{2} \cdot 105})}^{2} = y$

Step 2

Simplify each term.

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Step 2.1

Move

105

$105$ to the left of

\sqrt{2}

$\sqrt{2}$ .

x - 16 {(\frac{x}{105 \sqrt{2}})}^{2} = y

$x - 16 {(\frac{x}{105 \sqrt{2}})}^{2} = y$

Step 2.2

Multiply

\frac{x}{105 \sqrt{2}}

$\frac{x}{105 \sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

x - 16 {(\frac{x}{105 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2} = y

$x - 16 {(\frac{x}{105 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2} = y$

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

Multiply

\frac{x}{105 \sqrt{2}}

$\frac{x}{105 \sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

x - 16 {(\frac{x \sqrt{2}}{105 \sqrt{2} \sqrt{2}})}^{2} = y

$x - 16 {(\frac{x \sqrt{2}}{105 \sqrt{2} \sqrt{2}})}^{2} = y$

Step 2.3.2

Move

\sqrt{2}

$\sqrt{2}$ .

x - 16 {⎛ ⎜ ⎝ \frac{x \sqrt{2}}{105 (\sqrt{2} \sqrt{2})} ⎞ ⎟ ⎠}^{2} = y

$x - 16 {(\frac{x \sqrt{2}}{105 (\sqrt{2} \sqrt{2})})}^{2} = y$

Step 2.3.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

x - 16 {⎛ ⎜ ⎝ \frac{x \sqrt{2}}{105 ({\sqrt{2}}^{1} \sqrt{2})} ⎞ ⎟ ⎠}^{2} = y

$x - 16 {(\frac{x \sqrt{2}}{105 ({\sqrt{2}}^{1} \sqrt{2})})}^{2} = y$

Step 2.3.4

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

x - 16 {⎛ ⎜ ⎝ \frac{x \sqrt{2}}{105 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})} ⎞ ⎟ ⎠}^{2} = y

$x - 16 {(\frac{x \sqrt{2}}{105 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})})}^{2} = y$

Step 2.3.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

x - 16 {(\frac{x \sqrt{2}}{105 {\sqrt{2}}^{1 + 1}})}^{2} = y

$x - 16 {(\frac{x \sqrt{2}}{105 {\sqrt{2}}^{1 + 1}})}^{2} = y$

Step 2.3.6

Add

1

$1$ and

1

$1$ .

x - 16 {(\frac{x \sqrt{2}}{105 {\sqrt{2}}^{2}})}^{2} = y

$x - 16 {(\frac{x \sqrt{2}}{105 {\sqrt{2}}^{2}})}^{2} = y$

Step 2.3.7

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

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Step 2.3.7.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

x - 16 {⎛ ⎜ ⎜ ⎝ \frac{x \sqrt{2}}{105 {(2^{\frac{1}{2}})}^{2}} ⎞ ⎟ ⎟ ⎠}^{2} = y

$x - 16 {(\frac{x \sqrt{2}}{105 {(2^{\frac{1}{2}})}^{2}})}^{2} = y$

Step 2.3.7.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

x - 16 {(\frac{x \sqrt{2}}{105 \cdot 2^{\frac{1}{2} \cdot 2}})}^{2} = y

$x - 16 {(\frac{x \sqrt{2}}{105 \cdot 2^{\frac{1}{2} \cdot 2}})}^{2} = y$

Step 2.3.7.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

x - 16 {(\frac{x \sqrt{2}}{105 \cdot 2^{\frac{2}{2}}})}^{2} = y

$x - 16 {(\frac{x \sqrt{2}}{105 \cdot 2^{\frac{2}{2}}})}^{2} = y$

Step 2.3.7.4

Cancel the common factor of

2

$2$ .

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Step 2.3.7.4.1

Cancel the common factor.

$x - 16 {(\frac{x \sqrt{2}}{105 \cdot 2^{\frac{2}{2}}})}^{2} = y$

Step 2.3.7.4.2

Rewrite the expression.

$x - 16 {(\frac{x \sqrt{2}}{105 \cdot 2^{1}})}^{2} = y$

Step 2.3.7.5

Evaluate the exponent.

$x - 16 {(\frac{x \sqrt{2}}{105 \cdot 2})}^{2} = y$

Step 2.4

Multiply

$105$ by

$2$ .

$x - 16 {(\frac{x \sqrt{2}}{210})}^{2} = y$

Step 2.5

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.5.1

Apply the product rule to

$\frac{x \sqrt{2}}{210}$ .

$x - 16 \frac{{(x \sqrt{2})}^{2}}{210^{2}} = y$

Step 2.5.2

Apply the product rule to

$x \sqrt{2}$ .

$x - 16 \frac{x^{2} {\sqrt{2}}^{2}}{210^{2}} = y$

Step 2.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 2.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$x - 16 \frac{x^{2} {(2^{\frac{1}{2}})}^{2}}{210^{2}} = y$

Step 2.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$x - 16 \frac{x^{2} \cdot 2^{\frac{1}{2} \cdot 2}}{210^{2}} = y$

Step 2.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$x - 16 \frac{x^{2} \cdot 2^{\frac{2}{2}}}{210^{2}} = y$

Step 2.6.4

Cancel the common factor of

$2$ .

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Step 2.6.4.1

Cancel the common factor.

$x - 16 \frac{x^{2} \cdot 2^{\frac{2}{2}}}{210^{2}} = y$

Step 2.6.4.2

Rewrite the expression.

$x - 16 \frac{x^{2} \cdot 2^{1}}{210^{2}} = y$

Step 2.6.5

Evaluate the exponent.

$x - 16 \frac{x^{2} \cdot 2}{210^{2}} = y$

Step 2.7

Raise

$210$ to the power of

$2$ .

$x - 16 \frac{x^{2} \cdot 2}{44100} = y$

Step 2.8

Cancel the common factor of

$4$ .

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Step 2.8.1

Factor

$4$ out of

$- 16$ .

$x + 4 (- 4) \frac{x^{2} \cdot 2}{44100} = y$

Step 2.8.2

Factor

$4$ out of

$44100$ .

$x + 4 \cdot - 4 \frac{x^{2} \cdot 2}{4 \cdot 11025} = y$

Step 2.8.3

Cancel the common factor.

$x + 4 \cdot - 4 \frac{x^{2} \cdot 2}{4 \cdot 11025} = y$

Step 2.8.4

Rewrite the expression.

$x - 4 \frac{x^{2} \cdot 2}{11025} = y$

Step 2.9

Combine

$- 4$ and

$\frac{x^{2} \cdot 2}{11025}$ .

$x + \frac{- 4 (x^{2} \cdot 2)}{11025} = y$

Step 2.10

Multiply

$2$ by

$- 4$ .

$x + \frac{- 8 x^{2}}{11025} = y$

Step 2.11

Move the negative in front of the fraction.

$x - \frac{8 x^{2}}{11025} = y$

Step 3

Subtract

$y$ from both sides of the equation.

$x - \frac{8 x^{2}}{11025} - y = 0$

Step 4

Multiply through by the least common denominator

$11025$ , then simplify.

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Step 4.1

Apply the distributive property.

$11025 x + 11025 (- \frac{8 x^{2}}{11025}) + 11025 (- y) = 0$

Step 4.2

Simplify.

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Step 4.2.1

Cancel the common factor of

$11025$ .

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Step 4.2.1.1

Move the leading negative in

$- \frac{8 x^{2}}{11025}$ into the numerator.

$11025 x + 11025 (\frac{- 8 x^{2}}{11025}) + 11025 (- y) = 0$

Step 4.2.1.2

Cancel the common factor.

$11025 x + 11025 (\frac{- 8 x^{2}}{11025}) + 11025 (- y) = 0$

Step 4.2.1.3

Rewrite the expression.

$11025 x - 8 x^{2} + 11025 (- y) = 0$

Step 4.2.2

Multiply

$- 1$ by

$11025$ .

$11025 x - 8 x^{2} - 11025 y = 0$

Step 4.3

Move

$11025 x$ .

$- 8 x^{2} - 11025 y + 11025 x = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values

$a = - 8$ ,

$b = 11025$ , and

$c = - 11025 y$ into the quadratic formula and solve for

$x$ .

$\frac{- 11025 \pm \sqrt{11025^{2} - 4 \cdot (- 8 \cdot (- 11025 y))}}{2 \cdot - 8}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise

$11025$ to the power of

$2$ .

$x = \frac{- 11025 \pm \sqrt{121550625 - 4 \cdot - 8 \cdot (- 11025 y)}}{2 \cdot - 8}$

Step 7.1.2

Multiply

$- 4 \cdot - 8 \cdot - 11025$ .

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Step 7.1.2.1

Multiply

$- 4$ by

$- 8$ .

$x = \frac{- 11025 \pm \sqrt{121550625 + 32 \cdot (- 11025 y)}}{2 \cdot - 8}$

Step 7.1.2.2

Multiply

$32$ by

$- 11025$ .

$x = \frac{- 11025 \pm \sqrt{121550625 - 352800 y}}{2 \cdot - 8}$

Step 7.1.3

Factor

$11025$ out of

$121550625 - 352800 y$ .

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Step 7.1.3.1

Factor

$11025$ out of

$121550625$ .

$x = \frac{- 11025 \pm \sqrt{11025 (11025) - 352800 y}}{2 \cdot - 8}$

Step 7.1.3.2

Factor

$11025$ out of

$- 352800 y$ .

$x = \frac{- 11025 \pm \sqrt{11025 (11025) + 11025 (- 32 y)}}{2 \cdot - 8}$

Step 7.1.3.3

Factor

$11025$ out of

$11025 (11025) + 11025 (- 32 y)$ .

$x = \frac{- 11025 \pm \sqrt{11025 (11025 - 32 y)}}{2 \cdot - 8}$

Step 7.1.4

Rewrite

$11025 (11025 - 32 y)$ as

$105^{2} (105^{2} - 32 y)$ .

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Step 7.1.4.1

Rewrite

$11025$ as

$105^{2}$ .

$x = \frac{- 11025 \pm \sqrt{105^{2} (11025 - 32 y)}}{2 \cdot - 8}$

Step 7.1.4.2

Rewrite

$11025$ as

$105^{2}$ .

$x = \frac{- 11025 \pm \sqrt{105^{2} (105^{2} - 32 y)}}{2 \cdot - 8}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 11025 \pm 105 \sqrt{105^{2} - 32 y}}{2 \cdot - 8}$

Step 7.1.6

Raise

$105$ to the power of

$2$ .

$x = \frac{- 11025 \pm 105 \sqrt{11025 - 32 y}}{2 \cdot - 8}$

Step 7.2

Multiply

$2$ by

$- 8$ .

$x = \frac{- 11025 \pm 105 \sqrt{11025 - 32 y}}{- 16}$

Step 7.3

Simplify

$\frac{- 11025 \pm 105 \sqrt{11025 - 32 y}}{- 16}$ .

$x = \frac{11025 \pm 105 \sqrt{11025 - 32 y}}{16}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{105 (105 + \sqrt{11025 - 32 y})}{16}$

$x = \frac{105 (105 - \sqrt{11025 - 32 y})}{16}$