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sinh^2(x)+3tanh^2(x)=4

Solution

sinh^{2} (x) + 3 tanh^{2} (x) = 4

Solution

x = \frac{1}{2} ln (\frac{5.05103 E 15}{5.0 E 16}), x = \frac{1}{2} ln (\frac{4.94949 E 15}{5.0 E 14})

Degrees

x = - 65.67332 \dots^{\circ}, x = 65.67332 \dots^{\circ}

Solution steps

sinh^{2} (x) + 3 tanh^{2} (x) = 4

Rewrite using trig identities

sinh^{2} (x) + 3 tanh^{2} (x) = 4

Use the Hyperbolic identity:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

(\frac{e ^{x} - e ^{- x}}{2})^{2} + 3 tanh^{2} (x) = 4

Use the Hyperbolic identity:

tanh (x) = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

(\frac{e ^{x} - e ^{- x}}{2})^{2} + 3 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} = 4

(\frac{e ^{x} - e ^{- x}}{2})^{2} + 3 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} = 4

(\frac{e ^{x} - e ^{- x}}{2})^{2} + 3 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} = 4 : x = \frac{1}{2} ln (\frac{5.05103 E 15}{5.0 E 16}), x = \frac{1}{2} ln (\frac{4.94949 E 15}{5.0 E 14})

(\frac{e ^{x} - e ^{- x}}{2})^{2} + 3 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} = 4

Apply exponent rules

(\frac{e ^{x} - e ^{- x}}{2})^{2} + 3 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} = 4

Apply exponent rule:

a^{b c} = (a^{b})^{c}

e^{- x} = (e^{x})^{- 1}

(\frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2})^{2} + 3 (\frac{e ^{x} - ( e ^{x} ) ^{- 1}}{e ^{x} + ( e ^{x} ) ^{- 1}})^{2} = 4

(\frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2})^{2} + 3 (\frac{e ^{x} - ( e ^{x} ) ^{- 1}}{e ^{x} + ( e ^{x} ) ^{- 1}})^{2} = 4

Rewrite the equation with

e^{x} = u

(\frac{u - ( u ) ^{- 1}}{2})^{2} + 3 (\frac{u - ( u ) ^{- 1}}{u + ( u ) ^{- 1}})^{2} = 4

Solve

(\frac{u - u ^{- 1}}{2})^{2} + 3 (\frac{u - u ^{- 1}}{u + u ^{- 1}})^{2} = 4 : u = \frac{5.05103 E 15}{5.0 E 16}, u = - \frac{5.05103 E 15}{5.0 E 16}, u = \frac{4.94949 E 15}{5.0 E 14}, u = - \frac{4.94949 E 15}{5.0 E 14}

(\frac{u - u ^{- 1}}{2})^{2} + 3 (\frac{u - u ^{- 1}}{u + u ^{- 1}})^{2} = 4

Refine

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} + \frac{3 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} = 4

Multiply by LCM

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} + \frac{3 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} = 4

Find Least Common Multiplier of

4 u^{2}, (u^{2} + 1)^{2} : 4 u^{2} (u^{2} + 1)^{2}

4 u^{2}, (u^{2} + 1)^{2}

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

4 u^{2}

(u^{2} + 1)^{2}

= 4 u^{2} (u^{2} + 1)^{2}

Multiply by LCM=

4 u^{2} (u^{2} + 1)^{2}

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} (u^{2} + 1)^{2} + \frac{3 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} \cdot 4 u^{2} (u^{2} + 1)^{2} = 4 \cdot 4 u^{2} (u^{2} + 1)^{2}

Simplify

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} (u^{2} + 1)^{2} + \frac{3 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} \cdot 4 u^{2} (u^{2} + 1)^{2} = 4 \cdot 4 u^{2} (u^{2} + 1)^{2}

Simplify

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} (u^{2} + 1)^{2} : (u^{2} - 1)^{2} (u^{2} + 1)^{2}

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} (u^{2} + 1)^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( u ^{2} - 1 ) ^{2} \cdot 4 u ^{2} ( u ^{2} + 1 ) ^{2}}{4 u ^{2}}

Cancel the common factor:

4

= \frac{( u ^{2} - 1 ) ^{2} u ^{2} ( u ^{2} + 1 ) ^{2}}{u ^{2}}

Cancel the common factor:

u^{2}

= (u^{2} - 1)^{2} (u^{2} + 1)^{2}

Simplify

\frac{3 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} \cdot 4 u^{2} (u^{2} + 1)^{2} : 12 u^{2} (u^{2} - 1)^{2}

\frac{3 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} \cdot 4 u^{2} (u^{2} + 1)^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 ( u ^{2} - 1 ) ^{2} \cdot 4 u ^{2} ( u ^{2} + 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}}

Cancel the common factor:

(u^{2} + 1)^{2}

= 3 (u^{2} - 1)^{2} \cdot 4 u^{2}

Multiply the numbers:

3 \cdot 4 = 12

= 12 u^{2} (u^{2} - 1)^{2}

Simplify

4 \cdot 4 u^{2} (u^{2} + 1)^{2} : 16 u^{2} (u^{2} + 1)^{2}

4 \cdot 4 u^{2} (u^{2} + 1)^{2}

Multiply the numbers:

4 \cdot 4 = 16

= 16 u^{2} (u^{2} + 1)^{2}

(u^{2} - 1)^{2} (u^{2} + 1)^{2} + 12 u^{2} (u^{2} - 1)^{2} = 16 u^{2} (u^{2} + 1)^{2}

(u^{2} - 1)^{2} (u^{2} + 1)^{2} + 12 u^{2} (u^{2} - 1)^{2} = 16 u^{2} (u^{2} + 1)^{2}

(u^{2} - 1)^{2} (u^{2} + 1)^{2} + 12 u^{2} (u^{2} - 1)^{2} = 16 u^{2} (u^{2} + 1)^{2}

Solve

(u^{2} - 1)^{2} (u^{2} + 1)^{2} + 12 u^{2} (u^{2} - 1)^{2} = 16 u^{2} (u^{2} + 1)^{2} : u = \frac{5.05103 E 15}{5.0 E 16}, u = - \frac{5.05103 E 15}{5.0 E 16}, u = \frac{4.94949 E 15}{5.0 E 14}, u = - \frac{4.94949 E 15}{5.0 E 14}

(u^{2} - 1)^{2} (u^{2} + 1)^{2} + 12 u^{2} (u^{2} - 1)^{2} = 16 u^{2} (u^{2} + 1)^{2}

Expand

(u^{2} - 1)^{2} (u^{2} + 1)^{2} + 12 u^{2} (u^{2} - 1)^{2} : u^{8} + 12 u^{6} - 26 u^{4} + 12 u^{2} + 1

(u^{2} - 1)^{2} (u^{2} + 1)^{2} + 12 u^{2} (u^{2} - 1)^{2}

(u^{2} - 1)^{2} (u^{2} + 1)^{2} = (u^{4} - 2 u^{2} + 1) (u^{4} + 2 u^{2} + 1)

(u^{2} - 1)^{2} (u^{2} + 1)^{2}

(u^{2} - 1)^{2} = u^{4} - 2 u^{2} + 1

(u^{2} - 1)^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2}

Simplify

(u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2} : u^{4} - 2 u^{2} + 1

(u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} - 2 \cdot 1 \cdot u^{2} + 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

2 u^{2} \cdot 1 = 2 u^{2}

2 u^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

= u^{4} - 2 u^{2} + 1

= u^{4} - 2 u^{2} + 1

= (u^{4} - 2 u^{2} + 1) (u^{2} + 1)^{2}

(u^{2} + 1)^{2} = u^{4} + 2 u^{2} + 1

(u^{2} + 1)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

Simplify

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2} : u^{4} + 2 u^{2} + 1

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} + 2 \cdot 1 \cdot u^{2} + 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

2 u^{2} \cdot 1 = 2 u^{2}

2 u^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

= u^{4} + 2 u^{2} + 1

= u^{4} + 2 u^{2} + 1

= (u^{4} - 2 u^{2} + 1) (u^{4} + 2 u^{2} + 1)

12 u^{2} (u^{2} - 1)^{2} = 12 u^{2} (u^{4} - 2 u^{2} + 1)

12 u^{2} (u^{2} - 1)^{2}

(u^{2} - 1)^{2} = u^{4} - 2 u^{2} + 1

(u^{2} - 1)^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2}

Simplify

(u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2} : u^{4} - 2 u^{2} + 1

(u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} - 2 \cdot 1 \cdot u^{2} + 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

2 u^{2} \cdot 1 = 2 u^{2}

2 u^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

= u^{4} - 2 u^{2} + 1

= u^{4} - 2 u^{2} + 1

= 12 u^{2} (u^{4} - 2 u^{2} + 1)

= (u^{4} - 2 u^{2} + 1) (u^{4} + 2 u^{2} + 1) + 12 u^{2} (u^{4} - 2 u^{2} + 1)

Expand

(u^{4} - 2 u^{2} + 1) (u^{4} + 2 u^{2} + 1) : u^{8} - 2 u^{4} + 1

(u^{4} - 2 u^{2} + 1) (u^{4} + 2 u^{2} + 1)

Distribute parentheses

= u^{4} u^{4} + u^{4} \cdot 2 u^{2} + u^{4} \cdot 1 + (- 2 u^{2}) u^{4} + (- 2 u^{2}) \cdot 2 u^{2} + (- 2 u^{2}) \cdot 1 + 1 \cdot u^{4} + 1 \cdot 2 u^{2} + 1 \cdot 1

Apply minus-plus rules

+ (- a) = - a

= u^{4} u^{4} + 2 u^{4} u^{2} + 1 \cdot u^{4} - 2 u^{4} u^{2} - 2 \cdot 2 u^{2} u^{2} - 2 \cdot 1 \cdot u^{2} + 1 \cdot u^{4} + 1 \cdot 2 u^{2} + 1 \cdot 1

Simplify

u^{4} u^{4} + 2 u^{4} u^{2} + 1 \cdot u^{4} - 2 u^{4} u^{2} - 2 \cdot 2 u^{2} u^{2} - 2 \cdot 1 \cdot u^{2} + 1 \cdot u^{4} + 1 \cdot 2 u^{2} + 1 \cdot 1 : u^{8} - 2 u^{4} + 1

u^{4} u^{4} + 2 u^{4} u^{2} + 1 \cdot u^{4} - 2 u^{4} u^{2} - 2 \cdot 2 u^{2} u^{2} - 2 \cdot 1 \cdot u^{2} + 1 \cdot u^{4} + 1 \cdot 2 u^{2} + 1 \cdot 1

Group like terms

= u^{4} u^{4} + 2 u^{4} u^{2} + 1 \cdot u^{4} - 2 u^{4} u^{2} + 1 \cdot u^{4} - 2 \cdot 2 u^{2} u^{2} - 2 \cdot 1 \cdot u^{2} + 1 \cdot 2 u^{2} + 1 \cdot 1

Add similar elements:

2 u^{4} u^{2} - 2 u^{4} u^{2} = 0

= u^{4} u^{4} + 1 \cdot u^{4} + 1 \cdot u^{4} - 2 \cdot 2 u^{2} u^{2} - 2 \cdot 1 \cdot u^{2} + 1 \cdot 2 u^{2} + 1 \cdot 1

Add similar elements:

1 \cdot u^{4} + 1 \cdot u^{4} = 2 u^{4}

= u^{4} u^{4} + 2 u^{4} - 2 \cdot 2 u^{2} u^{2} - 2 \cdot 1 \cdot u^{2} + 1 \cdot 2 u^{2} + 1 \cdot 1

Add similar elements:

- 2 \cdot 1 \cdot u^{2} + 1 \cdot 2 u^{2} = 0

= u^{4} u^{4} + 2 u^{4} - 2 \cdot 2 u^{2} u^{2} + 1 \cdot 1

u^{4} u^{4} = u^{8}

u^{4} u^{4}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{4} u^{4} = u^{4 + 4}

= u^{4 + 4}

Add the numbers:

4 + 4 = 8

= u^{8}

2 \cdot 2 u^{2} u^{2} = 4 u^{4}

2 \cdot 2 u^{2} u^{2}

Multiply the numbers:

2 \cdot 2 = 4

= 4 u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= 4 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= 4 u^{4}

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

= u^{8} + 2 u^{4} - 4 u^{4} + 1

Add similar elements:

2 u^{4} - 4 u^{4} = - 2 u^{4}

= u^{8} - 2 u^{4} + 1

= u^{8} - 2 u^{4} + 1

= u^{8} - 2 u^{4} + 1 + 12 u^{2} (u^{4} - 2 u^{2} + 1)

Expand

12 u^{2} (u^{4} - 2 u^{2} + 1) : 12 u^{6} - 24 u^{4} + 12 u^{2}

12 u^{2} (u^{4} - 2 u^{2} + 1)

Distribute parentheses

= 12 u^{2} u^{4} + 12 u^{2} (- 2 u^{2}) + 12 u^{2} \cdot 1

Apply minus-plus rules

+ (- a) = - a

= 12 u^{4} u^{2} - 12 \cdot 2 u^{2} u^{2} + 12 \cdot 1 \cdot u^{2}

Simplify

12 u^{4} u^{2} - 12 \cdot 2 u^{2} u^{2} + 12 \cdot 1 \cdot u^{2} : 12 u^{6} - 24 u^{4} + 12 u^{2}

12 u^{4} u^{2} - 12 \cdot 2 u^{2} u^{2} + 12 \cdot 1 \cdot u^{2}

12 u^{4} u^{2} = 12 u^{6}

12 u^{4} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{4} u^{2} = u^{4 + 2}

= 12 u^{4 + 2}

Add the numbers:

4 + 2 = 6

= 12 u^{6}

12 \cdot 2 u^{2} u^{2} = 24 u^{4}

12 \cdot 2 u^{2} u^{2}

Multiply the numbers:

12 \cdot 2 = 24

= 24 u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= 24 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= 24 u^{4}

12 \cdot 1 \cdot u^{2} = 12 u^{2}

12 \cdot 1 \cdot u^{2}

Multiply the numbers:

12 \cdot 1 = 12

= 12 u^{2}

= 12 u^{6} - 24 u^{4} + 12 u^{2}

= 12 u^{6} - 24 u^{4} + 12 u^{2}

= u^{8} - 2 u^{4} + 1 + 12 u^{6} - 24 u^{4} + 12 u^{2}

Simplify

u^{8} - 2 u^{4} + 1 + 12 u^{6} - 24 u^{4} + 12 u^{2} : u^{8} + 12 u^{6} - 26 u^{4} + 12 u^{2} + 1

u^{8} - 2 u^{4} + 1 + 12 u^{6} - 24 u^{4} + 12 u^{2}

Group like terms

= u^{8} + 12 u^{6} - 2 u^{4} - 24 u^{4} + 12 u^{2} + 1

Add similar elements:

- 2 u^{4} - 24 u^{4} = - 26 u^{4}

= u^{8} + 12 u^{6} - 26 u^{4} + 12 u^{2} + 1

= u^{8} + 12 u^{6} - 26 u^{4} + 12 u^{2} + 1

Expand

16 u^{2} (u^{2} + 1)^{2} : 16 u^{6} + 32 u^{4} + 16 u^{2}

16 u^{2} (u^{2} + 1)^{2}

(u^{2} + 1)^{2} = u^{4} + 2 u^{2} + 1

(u^{2} + 1)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

Simplify

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2} : u^{4} + 2 u^{2} + 1

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} + 2 \cdot 1 \cdot u^{2} + 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

2 u^{2} \cdot 1 = 2 u^{2}

2 u^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

= u^{4} + 2 u^{2} + 1

= u^{4} + 2 u^{2} + 1

= 16 u^{2} (u^{4} + 2 u^{2} + 1)

Distribute parentheses

= 16 u^{2} u^{4} + 16 u^{2} \cdot 2 u^{2} + 16 u^{2} \cdot 1

= 16 u^{4} u^{2} + 16 \cdot 2 u^{2} u^{2} + 16 \cdot 1 \cdot u^{2}

Simplify

16 u^{4} u^{2} + 16 \cdot 2 u^{2} u^{2} + 16 \cdot 1 \cdot u^{2} : 16 u^{6} + 32 u^{4} + 16 u^{2}

16 u^{4} u^{2} + 16 \cdot 2 u^{2} u^{2} + 16 \cdot 1 \cdot u^{2}

16 u^{4} u^{2} = 16 u^{6}

16 u^{4} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{4} u^{2} = u^{4 + 2}

= 16 u^{4 + 2}

Add the numbers:

4 + 2 = 6

= 16 u^{6}

16 \cdot 2 u^{2} u^{2} = 32 u^{4}

16 \cdot 2 u^{2} u^{2}

Multiply the numbers:

16 \cdot 2 = 32

= 32 u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= 32 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= 32 u^{4}

16 \cdot 1 \cdot u^{2} = 16 u^{2}

16 \cdot 1 \cdot u^{2}

Multiply the numbers:

16 \cdot 1 = 16

= 16 u^{2}

= 16 u^{6} + 32 u^{4} + 16 u^{2}

= 16 u^{6} + 32 u^{4} + 16 u^{2}

u^{8} + 12 u^{6} - 26 u^{4} + 12 u^{2} + 1 = 16 u^{6} + 32 u^{4} + 16 u^{2}

Move

16 u^{2}

to the left side

u^{8} + 12 u^{6} - 26 u^{4} + 12 u^{2} + 1 = 16 u^{6} + 32 u^{4} + 16 u^{2}

Subtract

16 u^{2}

from both sides

u^{8} + 12 u^{6} - 26 u^{4} + 12 u^{2} + 1 - 16 u^{2} = 16 u^{6} + 32 u^{4} + 16 u^{2} - 16 u^{2}

Simplify

u^{8} + 12 u^{6} - 26 u^{4} - 4 u^{2} + 1 = 16 u^{6} + 32 u^{4}

u^{8} + 12 u^{6} - 26 u^{4} - 4 u^{2} + 1 = 16 u^{6} + 32 u^{4}

Move

32 u^{4}

to the left side

u^{8} + 12 u^{6} - 26 u^{4} - 4 u^{2} + 1 = 16 u^{6} + 32 u^{4}

Subtract

32 u^{4}

from both sides

u^{8} + 12 u^{6} - 26 u^{4} - 4 u^{2} + 1 - 32 u^{4} = 16 u^{6} + 32 u^{4} - 32 u^{4}

Simplify

u^{8} + 12 u^{6} - 58 u^{4} - 4 u^{2} + 1 = 16 u^{6}

u^{8} + 12 u^{6} - 58 u^{4} - 4 u^{2} + 1 = 16 u^{6}

Move

16 u^{6}

to the left side

u^{8} + 12 u^{6} - 58 u^{4} - 4 u^{2} + 1 = 16 u^{6}

Subtract

16 u^{6}

from both sides

u^{8} + 12 u^{6} - 58 u^{4} - 4 u^{2} + 1 - 16 u^{6} = 16 u^{6} - 16 u^{6}

Simplify

u^{8} - 4 u^{6} - 58 u^{4} - 4 u^{2} + 1 = 0

u^{8} - 4 u^{6} - 58 u^{4} - 4 u^{2} + 1 = 0

Rewrite the equation with

v = u^{2}, v^{2} = u^{4}, v^{3} = u^{6}

and

v^{4} = u^{8}

v^{4} - 4 v^{3} - 58 v^{2} - 4 v + 1 = 0

Solve

v^{4} - 4 v^{3} - 58 v^{2} - 4 v + 1 = 0 : v \approx 0.10102 \dots, v \approx - 0.17157 \dots, v \approx - 5.82842 \dots, v \approx 9.89897 \dots

v^{4} - 4 v^{3} - 58 v^{2} - 4 v + 1 = 0

Find one solution for

v^{4} - 4 v^{3} - 58 v^{2} - 4 v + 1 = 0

using Newton-Raphson:

v \approx 0.10102 \dots

v^{4} - 4 v^{3} - 58 v^{2} - 4 v + 1 = 0

Newton-Raphson Approximation Definition

f (v) = v^{4} - 4 v^{3} - 58 v^{2} - 4 v + 1

Find

f^{^{'}} (v) : 4 v^{3} - 12 v^{2} - 116 v - 4

\frac{d}{d v} (v^{4} - 4 v^{3} - 58 v^{2} - 4 v + 1)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d v} (v^{4}) - \frac{d}{d v} (4 v^{3}) - \frac{d}{d v} (58 v^{2}) - \frac{d}{d v} (4 v) + \frac{d}{d v} (1)

\frac{d}{d v} (v^{4}) = 4 v^{3}

\frac{d}{d v} (v^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 v^{4 - 1}

Simplify

= 4 v^{3}

\frac{d}{d v} (4 v^{3}) = 12 v^{2}

\frac{d}{d v} (4 v^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{d}{d v} (v^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 \cdot 3 v^{3 - 1}

Simplify

= 12 v^{2}

\frac{d}{d v} (58 v^{2}) = 116 v

\frac{d}{d v} (58 v^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 58 \frac{d}{d v} (v^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 58 \cdot 2 v^{2 - 1}

Simplify

= 116 v

\frac{d}{d v} (4 v) = 4

\frac{d}{d v} (4 v)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 4 \cdot 1

Simplify

= 4

\frac{d}{d v} (1) = 0

\frac{d}{d v} (1)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 4 v^{3} - 12 v^{2} - 116 v - 4 + 0

Simplify

= 4 v^{3} - 12 v^{2} - 116 v - 4

Let

v_{0} = 0

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = 0.25 : Δ v_{1} = 0.25

f (v_{0}) = 0^{4} - 4 \cdot 0^{3} - 58 \cdot 0^{2} - 4 \cdot 0 + 1 = 1

f^{^{'}} (v_{0}) = 4 \cdot 0^{3} - 12 \cdot 0^{2} - 116 \cdot 0 - 4 = - 4

v_{1} = 0.25

Δ v_{1} = ∣ 0.25 - 0 ∣ = 0.25

Δ v_{1} = 0.25

v_{2} = 0.14065 \dots : Δ v_{2} = 0.10934 \dots

f (v_{1}) = 0.2 5^{4} - 4 \cdot 0.2 5^{3} - 58 \cdot 0.2 5^{2} - 4 \cdot 0.25 + 1 = - 3.68359375

f^{^{'}} (v_{1}) = 4 \cdot 0.2 5^{3} - 12 \cdot 0.2 5^{2} - 116 \cdot 0.25 - 4 = - 33.6875

v_{2} = 0.14065 \dots

Δ v_{2} = ∣ 0.14065 \dots - 0.25 ∣ = 0.10934 \dots

Δ v_{2} = 0.10934 \dots

v_{3} = 0.10556 \dots : Δ v_{3} = 0.03508 \dots

f (v_{2}) = 0.14065 \dots^{4} - 4 \cdot 0.14065 \dots^{3} - 58 \cdot 0.14065 \dots^{2} - 4 \cdot 0.14065 \dots + 1 = - 0.72080 \dots

f^{^{'}} (v_{2}) = 4 \cdot 0.14065 \dots^{3} - 12 \cdot 0.14065 \dots^{2} - 116 \cdot 0.14065 \dots - 4 = - 20.54213 \dots

v_{3} = 0.10556 \dots

Δ v_{3} = ∣ 0.10556 \dots - 0.14065 \dots ∣ = 0.03508 \dots

Δ v_{3} = 0.03508 \dots

v_{4} = 0.10109 \dots : Δ v_{4} = 0.00446 \dots

f (v_{3}) = 0.10556 \dots^{4} - 4 \cdot 0.10556 \dots^{3} - 58 \cdot 0.10556 \dots^{2} - 4 \cdot 0.10556 \dots + 1 = - 0.07319 \dots

f^{^{'}} (v_{3}) = 4 \cdot 0.10556 \dots^{3} - 12 \cdot 0.10556 \dots^{2} - 116 \cdot 0.10556 \dots - 4 = - 16.37457 \dots

v_{4} = 0.10109 \dots

Δ v_{4} = ∣ 0.10109 \dots - 0.10556 \dots ∣ = 0.00446 \dots

Δ v_{4} = 0.00446 \dots

v_{5} = 0.10102 \dots : Δ v_{5} = 0.00007 \dots

f (v_{4}) = 0.10109 \dots^{4} - 4 \cdot 0.10109 \dots^{3} - 58 \cdot 0.10109 \dots^{2} - 4 \cdot 0.10109 \dots + 1 = - 0.00118 \dots

f^{^{'}} (v_{4}) = 4 \cdot 0.10109 \dots^{3} - 12 \cdot 0.10109 \dots^{2} - 116 \cdot 0.10109 \dots - 4 = - 15.84554 \dots

v_{5} = 0.10102 \dots

Δ v_{5} = ∣ 0.10102 \dots - 0.10109 \dots ∣ = 0.00007 \dots

Δ v_{5} = 0.00007 \dots

v_{6} = 0.10102 \dots : Δ v_{6} = 2.08017 E - 8

f (v_{5}) = 0.10102 \dots^{4} - 4 \cdot 0.10102 \dots^{3} - 58 \cdot 0.10102 \dots^{2} - 4 \cdot 0.10102 \dots + 1 = - 3.29431 E - 7

f^{^{'}} (v_{5}) = 4 \cdot 0.10102 \dots^{3} - 12 \cdot 0.10102 \dots^{2} - 116 \cdot 0.10102 \dots - 4 = - 15.83672 \dots

v_{6} = 0.10102 \dots

Δ v_{6} = ∣ 0.10102 \dots - 0.10102 \dots ∣ = 2.08017 E - 8

Δ v_{6} = 2.08017 E - 8

v \approx 0.10102 \dots

Apply long division:

\frac{v ^{4} - 4 v ^{3} - 58 v ^{2} - 4 v + 1}{v - 0.10102 \dots} = v^{3} - 3.89897 \dots v^{2} - 58.39387 \dots v - 9.89897 \dots

v^{3} - 3.89897 \dots v^{2} - 58.39387 \dots v - 9.89897 \dots \approx 0

Find one solution for

v^{3} - 3.89897 \dots v^{2} - 58.39387 \dots v - 9.89897 \dots = 0

using Newton-Raphson:

v \approx - 0.17157 \dots

v^{3} - 3.89897 \dots v^{2} - 58.39387 \dots v - 9.89897 \dots = 0

Newton-Raphson Approximation Definition

f (v) = v^{3} - 3.89897 \dots v^{2} - 58.39387 \dots v - 9.89897 \dots

Find

f^{^{'}} (v) : 3 v^{2} - 7.79795 \dots v - 58.39387 \dots

\frac{d}{d v} (v^{3} - 3.89897 \dots v^{2} - 58.39387 \dots v - 9.89897 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d v} (v^{3}) - \frac{d}{d v} (3.89897 \dots v^{2}) - \frac{d}{d v} (58.39387 \dots v) - \frac{d}{d v} (9.89897 \dots)

\frac{d}{d v} (v^{3}) = 3 v^{2}

\frac{d}{d v} (v^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 v^{3 - 1}

Simplify

= 3 v^{2}

\frac{d}{d v} (3.89897 \dots v^{2}) = 7.79795 \dots v

\frac{d}{d v} (3.89897 \dots v^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 3.89897 \dots \frac{d}{d v} (v^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3.89897 \dots \cdot 2 v^{2 - 1}

Simplify

= 7.79795 \dots v

\frac{d}{d v} (58.39387 \dots v) = 58.39387 \dots

\frac{d}{d v} (58.39387 \dots v)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 58.39387 \dots \frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 58.39387 \dots \cdot 1

Simplify

= 58.39387 \dots

\frac{d}{d v} (9.89897 \dots) = 0

\frac{d}{d v} (9.89897 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 3 v^{2} - 7.79795 \dots v - 58.39387 \dots - 0

Simplify

= 3 v^{2} - 7.79795 \dots v - 58.39387 \dots

Let

v_{0} = 0

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = - 0.16952 \dots : Δ v_{1} = 0.16952 \dots

f (v_{0}) = 0^{3} - 3.89897 \dots \cdot 0^{2} - 58.39387 \dots \cdot 0 - 9.89897 \dots = - 9.89897 \dots

f^{^{'}} (v_{0}) = 3 \cdot 0^{2} - 7.79795 \dots \cdot 0 - 58.39387 \dots = - 58.39387 \dots

v_{1} = - 0.16952 \dots

Δ v_{1} = ∣ - 0.16952 \dots - 0 ∣ = 0.16952 \dots

Δ v_{1} = 0.16952 \dots

v_{2} = - 0.17157 \dots : Δ v_{2} = 0.00205 \dots

f (v_{1}) = (- 0.16952 \dots)^{3} - 3.89897 \dots (- 0.16952 \dots)^{2} - 58.39387 \dots (- 0.16952 \dots) - 9.89897 \dots = - 0.11691 \dots

f^{^{'}} (v_{1}) = 3 (- 0.16952 \dots)^{2} - 7.79795 \dots (- 0.16952 \dots) - 58.39387 \dots = - 56.98574 \dots

v_{2} = - 0.17157 \dots

Δ v_{2} = ∣ - 0.17157 \dots - (- 0.16952 \dots) ∣ = 0.00205 \dots

Δ v_{2} = 0.00205 \dots

v_{3} = - 0.17157 \dots : Δ v_{3} = 3.25836 E - 7

f (v_{2}) = (- 0.17157 \dots)^{3} - 3.89897 \dots (- 0.17157 \dots)^{2} - 58.39387 \dots (- 0.17157 \dots) - 9.89897 \dots = - 0.00001 \dots

f^{^{'}} (v_{2}) = 3 (- 0.17157 \dots)^{2} - 7.79795 \dots (- 0.17157 \dots) - 58.39387 \dots = - 56.96764 \dots

v_{3} = - 0.17157 \dots

Δ v_{3} = ∣ - 0.17157 \dots - (- 0.17157 \dots) ∣ = 3.25836 E - 7

Δ v_{3} = 3.25836 E - 7

v \approx - 0.17157 \dots

Apply long division:

\frac{v ^{3} - 3.89897 \dots v ^{2} - 58.39387 \dots v - 9.89897 \dots}{v + 0.17157 \dots} = v^{2} - 4.07055 \dots v - 57.69548 \dots

v^{2} - 4.07055 \dots v - 57.69548 \dots \approx 0

Find one solution for

v^{2} - 4.07055 \dots v - 57.69548 \dots = 0

using Newton-Raphson:

v \approx - 5.82842 \dots

v^{2} - 4.07055 \dots v - 57.69548 \dots = 0

Newton-Raphson Approximation Definition

f (v) = v^{2} - 4.07055 \dots v - 57.69548 \dots

Find

f^{^{'}} (v) : 2 v - 4.07055 \dots

\frac{d}{d v} (v^{2} - 4.07055 \dots v - 57.69548 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d v} (v^{2}) - \frac{d}{d v} (4.07055 \dots v) - \frac{d}{d v} (57.69548 \dots)

\frac{d}{d v} (v^{2}) = 2 v

\frac{d}{d v} (v^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 v^{2 - 1}

Simplify

= 2 v

\frac{d}{d v} (4.07055 \dots v) = 4.07055 \dots

\frac{d}{d v} (4.07055 \dots v)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4.07055 \dots \frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 4.07055 \dots \cdot 1

Simplify

= 4.07055 \dots

\frac{d}{d v} (57.69548 \dots) = 0

\frac{d}{d v} (57.69548 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 2 v - 4.07055 \dots - 0

Simplify

= 2 v - 4.07055 \dots

Let

v_{0} = - 5

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = - 5.87720 \dots : Δ v_{1} = 0.87720 \dots

f (v_{0}) = (- 5)^{2} - 4.07055 \dots (- 5) - 57.69548 \dots = - 12.34271 \dots

f^{^{'}} (v_{0}) = 2 (- 5) - 4.07055 \dots = - 14.07055 \dots

v_{1} = - 5.87720 \dots

Δ v_{1} = ∣ - 5.87720 \dots - (- 5) ∣ = 0.87720 \dots

Δ v_{1} = 0.87720 \dots

v_{2} = - 5.82857 \dots : Δ v_{2} = 0.04862 \dots

f (v_{1}) = (- 5.87720 \dots)^{2} - 4.07055 \dots (- 5.87720 \dots) - 57.69548 \dots = 0.76948 \dots

f^{^{'}} (v_{1}) = 2 (- 5.87720 \dots) - 4.07055 \dots = - 15.82495 \dots

v_{2} = - 5.82857 \dots

Δ v_{2} = ∣ - 5.82857 \dots - (- 5.87720 \dots) ∣ = 0.04862 \dots

Δ v_{2} = 0.04862 \dots

v_{3} = - 5.82842 \dots : Δ v_{3} = 0.00015 \dots

f (v_{2}) = (- 5.82857 \dots)^{2} - 4.07055 \dots (- 5.82857 \dots) - 57.69548 \dots = 0.00236 \dots

f^{^{'}} (v_{2}) = 2 (- 5.82857 \dots) - 4.07055 \dots = - 15.72770 \dots

v_{3} = - 5.82842 \dots

Δ v_{3} = ∣ - 5.82842 \dots - (- 5.82857 \dots) ∣ = 0.00015 \dots

Δ v_{3} = 0.00015 \dots

v_{4} = - 5.82842 \dots : Δ v_{4} = 1.43694 E - 9

f (v_{3}) = (- 5.82842 \dots)^{2} - 4.07055 \dots (- 5.82842 \dots) - 57.69548 \dots = 2.25994 E - 8

f^{^{'}} (v_{3}) = 2 (- 5.82842 \dots) - 4.07055 \dots = - 15.72740 \dots

v_{4} = - 5.82842 \dots

Δ v_{4} = ∣ - 5.82842 \dots - (- 5.82842 \dots) ∣ = 1.43694 E - 9

Δ v_{4} = 1.43694 E - 9

v \approx - 5.82842 \dots

Apply long division:

\frac{v ^{2} - 4.07055 \dots v - 57.69548 \dots}{v + 5.82842 \dots} = v - 9.89897 \dots

v - 9.89897 \dots \approx 0

v \approx 9.89897 \dots

The solutions are

v \approx 0.10102 \dots, v \approx - 0.17157 \dots, v \approx - 5.82842 \dots, v \approx 9.89897 \dots

v \approx 0.10102 \dots, v \approx - 0.17157 \dots, v \approx - 5.82842 \dots, v \approx 9.89897 \dots

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 0.10102 \dots : u = \frac{5.05103 E 15}{5.0 E 16}, u = - \frac{5.05103 E 15}{5.0 E 16}

u^{2} = 0.10102 \dots

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 0.10102 \dots, u = - 0.10102 \dots

0.10102 \dots = \frac{5.05103 E 15}{5.0 E 16}

0.10102 \dots

Multiply and divide by 10 for every number after the decimal point.

There are

17

digits to the right of the decimal point, therefore multiply and divide by

2147483647

= \frac{1.0 E 17 \cdot 0.10102 \dots}{1.0 E 17}

Multiply the numbers:

1.0 E 17 \cdot 0.10102 \dots = 1.01021 E 16

= \frac{1.01021 E 16}{1.0 E 17}

Cancel

\frac{1.01021 E 16}{1.0 E 17} : \frac{5.05103 E 15}{5.0 E 16}

\frac{1.01021 E 16}{1.0 E 17}

Factor the number:

1.01021 E 16 = 2 \cdot 5.05103 E 15

= \frac{2 \cdot 5.05103 E 15}{1.0 E 17}

Factor the number:

1.0 E 17 = 2 \cdot 5.0 E 16

= \frac{2 \cdot 5.05103 E 15}{2 \cdot 5.0 E 16}

Cancel the common factor:

2

= \frac{5.05103 E 15}{5.0 E 16}

= \frac{5.05103 E 15}{5.0 E 16}

- 0.10102 \dots = - \frac{5.05103 E 15}{5.0 E 16}

- 0.10102 \dots

Multiply and divide by 10 for every number after the decimal point.

There are

17

digits to the right of the decimal point, therefore multiply and divide by

2147483647

= - \frac{1.0 E 17 \cdot 0.10102 \dots}{1.0 E 17}

Multiply the numbers:

1.0 E 17 \cdot 0.10102 \dots = 1.01021 E 16

= - \frac{1.01021 E 16}{1.0 E 17}

Cancel

\frac{1.01021 E 16}{1.0 E 17} : \frac{5.05103 E 15}{5.0 E 16}

\frac{1.01021 E 16}{1.0 E 17}

Factor the number:

1.01021 E 16 = 2 \cdot 5.05103 E 15

= \frac{2 \cdot 5.05103 E 15}{1.0 E 17}

Factor the number:

1.0 E 17 = 2 \cdot 5.0 E 16

= \frac{2 \cdot 5.05103 E 15}{2 \cdot 5.0 E 16}

Cancel the common factor:

2

= \frac{5.05103 E 15}{5.0 E 16}

= - \frac{5.05103 E 15}{5.0 E 16}

u = \frac{5.05103 E 15}{5.0 E 16}, u = - \frac{5.05103 E 15}{5.0 E 16}

Solve

u^{2} = - 0.17157 \dots :

No Solution for

u \in R

u^{2} = - 0.17157 \dots

x^{2}

cannot be negative for

x \in R

No Solution for u \in R

Solve

u^{2} = - 5.82842 \dots :

No Solution for

u \in R

u^{2} = - 5.82842 \dots

x^{2}

cannot be negative for

x \in R

No Solution for u \in R

Solve

u^{2} = 9.89897 \dots : u = \frac{4.94949 E 15}{5.0 E 14}, u = - \frac{4.94949 E 15}{5.0 E 14}

u^{2} = 9.89897 \dots

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 9.89897 \dots, u = - 9.89897 \dots

9.89897 \dots = \frac{4.94949 E 15}{5.0 E 14}

9.89897 \dots

Multiply and divide by 10 for every number after the decimal point.

There are

15

digits to the right of the decimal point, therefore multiply and divide by

2147483647

= \frac{1.0 E 15 \cdot 9.89897 \dots}{1.0 E 15}

Multiply the numbers:

1.0 E 15 \cdot 9.89897 \dots = 9.89898 E 15

= \frac{9.89898 E 15}{1.0 E 15}

Cancel

\frac{9.89898 E 15}{1.0 E 15} : \frac{4.94949 E 15}{5.0 E 14}

\frac{9.89898 E 15}{1.0 E 15}

Factor the number:

9.89898 E 15 = 2 \cdot 4.94949 E 15

= \frac{2 \cdot 4.94949 E 15}{1.0 E 15}

Factor the number:

1.0 E 15 = 2 \cdot 5.0 E 14

= \frac{2 \cdot 4.94949 E 15}{2 \cdot 5.0 E 14}

Cancel the common factor:

2

= \frac{4.94949 E 15}{5.0 E 14}

= \frac{4.94949 E 15}{5.0 E 14}

- 9.89897 \dots = - \frac{4.94949 E 15}{5.0 E 14}

- 9.89897 \dots

Multiply and divide by 10 for every number after the decimal point.

There are

15

digits to the right of the decimal point, therefore multiply and divide by

2147483647

= - \frac{1.0 E 15 \cdot 9.89897 \dots}{1.0 E 15}

Multiply the numbers:

1.0 E 15 \cdot 9.89897 \dots = 9.89898 E 15

= - \frac{9.89898 E 15}{1.0 E 15}

Cancel

\frac{9.89898 E 15}{1.0 E 15} : \frac{4.94949 E 15}{5.0 E 14}

\frac{9.89898 E 15}{1.0 E 15}

Factor the number:

9.89898 E 15 = 2 \cdot 4.94949 E 15

= \frac{2 \cdot 4.94949 E 15}{1.0 E 15}

Factor the number:

1.0 E 15 = 2 \cdot 5.0 E 14

= \frac{2 \cdot 4.94949 E 15}{2 \cdot 5.0 E 14}

Cancel the common factor:

2

= \frac{4.94949 E 15}{5.0 E 14}

= - \frac{4.94949 E 15}{5.0 E 14}

u = \frac{4.94949 E 15}{5.0 E 14}, u = - \frac{4.94949 E 15}{5.0 E 14}

The solutions are

u = \frac{5.05103 E 15}{5.0 E 16}, u = - \frac{5.05103 E 15}{5.0 E 16}, u = \frac{4.94949 E 15}{5.0 E 14}, u = - \frac{4.94949 E 15}{5.0 E 14}

u = \frac{5.05103 E 15}{5.0 E 16}, u = - \frac{5.05103 E 15}{5.0 E 16}, u = \frac{4.94949 E 15}{5.0 E 14}, u = - \frac{4.94949 E 15}{5.0 E 14}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(\frac{u - u ^{- 1}}{2})^{2} + 3 (\frac{u - u ^{- 1}}{u + u ^{- 1}})^{2}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{5.05103 E 15}{5.0 E 16}, u = - \frac{5.05103 E 15}{5.0 E 16}, u = \frac{4.94949 E 15}{5.0 E 14}, u = - \frac{4.94949 E 15}{5.0 E 14}

u = \frac{5.05103 E 15}{5.0 E 16}, u = - \frac{5.05103 E 15}{5.0 E 16}, u = \frac{4.94949 E 15}{5.0 E 14}, u = - \frac{4.94949 E 15}{5.0 E 14}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{5.05103 E 15}{5.0 E 16} : x = \frac{1}{2} ln (\frac{5.05103 E 15}{5.0 E 16})

e^{x} = \frac{5.05103 E 15}{5.0 E 16}

Apply exponent rules

e^{x} = \frac{5.05103 E 15}{5.0 E 16}

Apply exponent rule:

a = a^{\frac{1}{2}}

\frac{5.05103 E 15}{5.0 E 16} = (\frac{5.05103 E 15}{5.0 E 16})^{\frac{1}{2}}

e^{x} = (\frac{5.05103 E 15}{5.0 E 16})^{\frac{1}{2}}

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln ((\frac{5.05103 E 15}{5.0 E 16})^{\frac{1}{2}})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln ((\frac{5.05103 E 15}{5.0 E 16})^{\frac{1}{2}})

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln ((\frac{5.05103 E 15}{5.0 E 16})^{\frac{1}{2}}) = \frac{1}{2} ln (\frac{5.05103 E 15}{5.0 E 16})

x = \frac{1}{2} ln (\frac{5.05103 E 15}{5.0 E 16})

x = \frac{1}{2} ln (\frac{5.05103 E 15}{5.0 E 16})

Solve

e^{x} = - \frac{5.05103 E 15}{5.0 E 16} :

No Solution for

x \in R

e^{x} = - \frac{5.05103 E 15}{5.0 E 16}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

Solve

e^{x} = \frac{4.94949 E 15}{5.0 E 14} : x = \frac{1}{2} ln (\frac{4.94949 E 15}{5.0 E 14})

e^{x} = \frac{4.94949 E 15}{5.0 E 14}

Apply exponent rules

e^{x} = \frac{4.94949 E 15}{5.0 E 14}

Apply exponent rule:

a = a^{\frac{1}{2}}

\frac{4.94949 E 15}{5.0 E 14} = (\frac{4.94949 E 15}{5.0 E 14})^{\frac{1}{2}}

e^{x} = (\frac{4.94949 E 15}{5.0 E 14})^{\frac{1}{2}}

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln ((\frac{4.94949 E 15}{5.0 E 14})^{\frac{1}{2}})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln ((\frac{4.94949 E 15}{5.0 E 14})^{\frac{1}{2}})

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln ((\frac{4.94949 E 15}{5.0 E 14})^{\frac{1}{2}}) = \frac{1}{2} ln (\frac{4.94949 E 15}{5.0 E 14})

x = \frac{1}{2} ln (\frac{4.94949 E 15}{5.0 E 14})

x = \frac{1}{2} ln (\frac{4.94949 E 15}{5.0 E 14})

Solve

e^{x} = - \frac{4.94949 E 15}{5.0 E 14} :

No Solution for

x \in R

e^{x} = - \frac{4.94949 E 15}{5.0 E 14}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = \frac{1}{2} ln (\frac{5.05103 E 15}{5.0 E 16}), x = \frac{1}{2} ln (\frac{4.94949 E 15}{5.0 E 14})

x = \frac{1}{2} ln (\frac{5.05103 E 15}{5.0 E 16}), x = \frac{1}{2} ln (\frac{4.94949 E 15}{5.0 E 14})

Graph

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\frac{sin ( 5 1 ^{\circ} )}{18} = \frac{sin ( c )}{17}

0= 1/2 cos(8sqrt(2)t)+1/8 sin(8sqrt(2)t)

0 = \frac{1}{2} cos (82 t) + \frac{1}{8} sin (82 t)

tan(θ)=-15/8

tan (θ) = - \frac{15}{8}

3cos(x)cot(x)+7=5csc(x)

3 cos (x) cot (x) + 7 = 5 csc (x)

-10sin(2x)=0

- 10 sin (2 x) = 0