Basic Math Examples

$100^{2} + {(\frac{h}{\tan (22)})}^{2} = {(\frac{h}{\tan (19)})}^{2}$

Step 1

Subtract ${(\frac{h}{\tan (19)})}^{2}$ from both sides of the equation.

$100^{2} + {(\frac{h}{\tan (22)})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2

Simplify the left side of the equation.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $100$ to the power of $2$ .

$10000 + {(\frac{h}{\tan (22)})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.2

Separate fractions.

$10000 + {(\frac{h}{1} \cdot \frac{1}{\tan (22)})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.3

Rewrite $\tan (22)$ in terms of sines and cosines.

$10000 + {(\frac{h}{1} \cdot \frac{1}{\frac{\sin (22)}{\cos (22)}})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.4

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (22)}{\cos (22)}$ .

$10000 + {(\frac{h}{1} \cdot \frac{\cos (22)}{\sin (22)})}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.5

Convert from $\frac{\cos (22)}{\sin (22)}$ to $\cot (22)$ .

$10000 + {(\frac{h}{1} \cdot \cot (22))}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.6

Divide $h$ by $1$ .

$10000 + {(h \cot (22))}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.7

Evaluate $\cot (22)$ .

$10000 + {(h \cdot 2.47508685)}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.8

Move $2.47508685$ to the left of $h$ .

$10000 + {(2.47508685 \cdot h)}^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.9

Apply the product rule to $2.47508685 h$ .

$10000 + {2.47508685}^{2} h^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.10

Raise $2.47508685$ to the power of $2$ .

$10000 + 6.12605493 h^{2} - {(\frac{h}{\tan (19)})}^{2} = 0$

Step 2.1.11

Separate fractions.

$10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \frac{1}{\tan (19)})}^{2} = 0$

Step 2.1.12

Rewrite $\tan (19)$ in terms of sines and cosines.

$10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \frac{1}{\frac{\sin (19)}{\cos (19)}})}^{2} = 0$

Step 2.1.13

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (19)}{\cos (19)}$ .

$10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \frac{\cos (19)}{\sin (19)})}^{2} = 0$

Step 2.1.14

Convert from $\frac{\cos (19)}{\sin (19)}$ to $\cot (19)$ .

$10000 + 6.12605493 h^{2} - {(\frac{h}{1} \cdot \cot (19))}^{2} = 0$

Step 2.1.15

Divide $h$ by $1$ .

$10000 + 6.12605493 h^{2} - {(h \cot (19))}^{2} = 0$

Step 2.1.16

Evaluate $\cot (19)$ .

$10000 + 6.12605493 h^{2} - {(h \cdot 2.90421087)}^{2} = 0$

Step 2.1.17

Move $2.90421087$ to the left of $h$ .

$10000 + 6.12605493 h^{2} - {(2.90421087 \cdot h)}^{2} = 0$

Step 2.1.18

Apply the product rule to $2.90421087 h$ .

$10000 + 6.12605493 h^{2} - ({2.90421087}^{2} h^{2}) = 0$

Step 2.1.19

Raise $2.90421087$ to the power of $2$ .

$10000 + 6.12605493 h^{2} - (8.43444082 h^{2}) = 0$

Step 2.1.20

Multiply $8.43444082$ by $- 1$ .

$10000 + 6.12605493 h^{2} - 8.43444082 h^{2} = 0$

Step 2.2

Subtract $8.43444082 h^{2}$ from $6.12605493 h^{2}$ .

$10000 - 2.30838589 h^{2} = 0$

Step 3

Factor $10000 - 2.30838589 h^{2}$ .

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Step 3.1

Rewrite $10000$ as $100^{2}$ .

$100^{2} - 2.30838589 h^{2} = 0$

Step 3.2

Rewrite $2.30838589 h^{2}$ as ${(1.51933731 h)}^{2}$ .

$100^{2} - {(1.51933731 h)}^{2} = 0$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 100$ and $b = 1.51933731 h$ .

$(100 + 1.51933731 h) (100 - (1.51933731 h)) = 0$

Step 3.4

Multiply $1.51933731$ by $- 1$ .

$(100 + 1.51933731 h) (100 - 1.51933731 h) = 0$

Step 4

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

$100 + 1.51933731 h = 0$

$100 - 1.51933731 h = 0$

Step 5

Set $100 + 1.51933731 h$ equal to $0$ and solve for $h$ .

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Step 5.1

Set $100 + 1.51933731 h$ equal to $0$ .

$100 + 1.51933731 h = 0$

Step 5.2

Solve $100 + 1.51933731 h = 0$ for $h$ .

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Step 5.2.1

Subtract $100$ from both sides of the equation.

$1.51933731 h = - 100$

Step 5.2.2

Divide each term in $1.51933731 h = - 100$ by $1.51933731$ and simplify.

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Step 5.2.2.1

Divide each term in $1.51933731 h = - 100$ by $1.51933731$ .

$\frac{1.51933731 h}{1.51933731} = \frac{- 100}{1.51933731}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $1.51933731$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{1.51933731 h}{1.51933731} = \frac{- 100}{1.51933731}$

Step 5.2.2.2.1.2

Divide $h$ by $1$ .

$h = \frac{- 100}{1.51933731}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Divide $- 100$ by $1.51933731$ .

$h = - 65.8181687$

Step 6

Set $100 - 1.51933731 h$ equal to $0$ and solve for $h$ .

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Step 6.1

Set $100 - 1.51933731 h$ equal to $0$ .

$100 - 1.51933731 h = 0$

Step 6.2

Solve $100 - 1.51933731 h = 0$ for $h$ .

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Step 6.2.1

Subtract $100$ from both sides of the equation.

$- 1.51933731 h = - 100$

Step 6.2.2

Divide each term in $- 1.51933731 h = - 100$ by $- 1.51933731$ and simplify.

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Step 6.2.2.1

Divide each term in $- 1.51933731 h = - 100$ by $- 1.51933731$ .

$\frac{- 1.51933731 h}{- 1.51933731} = \frac{- 100}{- 1.51933731}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $- 1.51933731$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{- 1.51933731 h}{- 1.51933731} = \frac{- 100}{- 1.51933731}$

Step 6.2.2.2.1.2

Divide $h$ by $1$ .

$h = \frac{- 100}{- 1.51933731}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Divide $- 100$ by $- 1.51933731$ .

$h = 65.8181687$

Step 7

The final solution is all the values that make $(100 + 1.51933731 h) (100 - 1.51933731 h) = 0$ true.

$h = - 65.8181687, 65.8181687$