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1/2 sinh(2x)-4/5 cosh(2x)+1=0

Solution

\frac{1}{2} sinh (2 x) - \frac{4}{5} cosh (2 x) + 1 = 0

Solution

x = \frac{1}{2} ln (\frac{10 - 61}{3}), x = \frac{1}{2} ln (\frac{10 + 61}{3})

Degrees

x = - 9.01906 \dots^{\circ}, x = 51.02652 \dots^{\circ}

Solution steps

\frac{1}{2} sinh (2 x) - \frac{4}{5} cosh (2 x) + 1 = 0

Rewrite using trig identities

\frac{1}{2} sinh (2 x) - \frac{4}{5} cosh (2 x) + 1 = 0

Use the Hyperbolic identity:

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

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} cosh (2 x) + 1 = 0

Use the Hyperbolic identity:

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

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 1 = 0

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 1 = 0

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 1 = 0 : x = \frac{1}{2} ln (\frac{10 - 61}{3}), x = \frac{1}{2} ln (\frac{10 + 61}{3})

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 1 = 0

Find Least Common Multiplier of

4, 10 : 20

4, 10

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

10 : 2 \cdot 5

10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 5

Multiply each factor the greatest number of times it occurs in either

4

10

= 2 \cdot 2 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 5 = 20

= 20

Multiply by LCM=

20

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} \cdot 20 - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} \cdot 20 + 1 \cdot 20 = 0 \cdot 20

Simplify

5 (e^{2 x} - e^{- 2 x}) - 8 (e^{2 x} + e^{- 2 x}) + 20 = 0

Apply exponent rules

5 (e^{2 x} - e^{- 2 x}) - 8 (e^{2 x} + e^{- 2 x}) + 20 = 0

Apply exponent rule:

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

e^{2 x} = (e^{x})^{2}, e^{- 2 x} = (e^{x})^{- 2}

5 ((e^{x})^{2} - (e^{x})^{- 2}) - 8 ((e^{x})^{2} + (e^{x})^{- 2}) + 20 = 0

5 ((e^{x})^{2} - (e^{x})^{- 2}) - 8 ((e^{x})^{2} + (e^{x})^{- 2}) + 20 = 0

Rewrite the equation with

e^{x} = u

5 ((u)^{2} - (u)^{- 2}) - 8 ((u)^{2} + (u)^{- 2}) + 20 = 0

Solve

5 (u^{2} - u^{- 2}) - 8 (u^{2} + u^{- 2}) + 20 = 0 : u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

5 (u^{2} - u^{- 2}) - 8 (u^{2} + u^{- 2}) + 20 = 0

Refine

5 (u^{2} - \frac{1}{u ^{2}}) - 8 (u^{2} + \frac{1}{u ^{2}}) + 20 = 0

Expand

5 (u^{2} - \frac{1}{u ^{2}}) - 8 (u^{2} + \frac{1}{u ^{2}}) + 20 : - 3 u^{2} - \frac{13}{u ^{2}} + 20

5 (u^{2} - \frac{1}{u ^{2}}) - 8 (u^{2} + \frac{1}{u ^{2}}) + 20

Expand

5 (u^{2} - \frac{1}{u ^{2}}) : 5 u^{2} - \frac{5}{u ^{2}}

5 (u^{2} - \frac{1}{u ^{2}})

Apply the distributive law:

a (b - c) = ab - a c

a = 5, b = u^{2}, c = \frac{1}{u ^{2}}

= 5 u^{2} - 5 \cdot \frac{1}{u ^{2}}

5 \cdot \frac{1}{u ^{2}} = \frac{5}{u ^{2}}

5 \cdot \frac{1}{u ^{2}}

Multiply fractions:

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

= \frac{1 \cdot 5}{u ^{2}}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{u ^{2}}

= 5 u^{2} - \frac{5}{u ^{2}}

= 5 u^{2} - \frac{5}{u ^{2}} - 8 (u^{2} + \frac{1}{u ^{2}}) + 20

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = - 8, b = u^{2}, c = \frac{1}{u ^{2}}

= - 8 u^{2} + (- 8) \frac{1}{u ^{2}}

Apply minus-plus rules

+ (- a) = - a

= - 8 u^{2} - 8 \cdot \frac{1}{u ^{2}}

8 \cdot \frac{1}{u ^{2}} = \frac{8}{u ^{2}}

8 \cdot \frac{1}{u ^{2}}

Multiply fractions:

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

= \frac{1 \cdot 8}{u ^{2}}

Multiply the numbers:

1 \cdot 8 = 8

= \frac{8}{u ^{2}}

= - 8 u^{2} - \frac{8}{u ^{2}}

= 5 u^{2} - \frac{5}{u ^{2}} - 8 u^{2} - \frac{8}{u ^{2}} + 20

Simplify

5 u^{2} - \frac{5}{u ^{2}} - 8 u^{2} - \frac{8}{u ^{2}} + 20 : - 3 u^{2} - \frac{13}{u ^{2}} + 20

5 u^{2} - \frac{5}{u ^{2}} - 8 u^{2} - \frac{8}{u ^{2}} + 20

Group like terms

= 5 u^{2} - 8 u^{2} - \frac{5}{u ^{2}} - \frac{8}{u ^{2}} + 20

Combine the fractions

- \frac{5}{u ^{2}} - \frac{8}{u ^{2}} : - \frac{13}{u ^{2}}

Apply rule

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

= \frac{- 5 - 8}{u ^{2}}

Subtract the numbers:

- 5 - 8 = - 13

= \frac{- 13}{u ^{2}}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{13}{u ^{2}}

= 5 u^{2} - 8 u^{2} - \frac{13}{u ^{2}} + 20

Add similar elements:

5 u^{2} - 8 u^{2} = - 3 u^{2}

= - 3 u^{2} - \frac{13}{u ^{2}} + 20

= - 3 u^{2} - \frac{13}{u ^{2}} + 20

- 3 u^{2} - \frac{13}{u ^{2}} + 20 = 0

Multiply both sides by

u^{2}

- 3 u^{2} - \frac{13}{u ^{2}} + 20 = 0

Multiply both sides by

u^{2}

- 3 u^{2} u^{2} - \frac{13}{u ^{2}} u^{2} + 20 u^{2} = 0 \cdot u^{2}

Simplify

- 3 u^{2} u^{2} - \frac{13}{u ^{2}} u^{2} + 20 u^{2} = 0 \cdot u^{2}

Simplify

- 3 u^{2} u^{2} : - 3 u^{4}

- 3 u^{2} u^{2}

Apply exponent rule:

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

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

= - 3 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 3 u^{4}

Simplify

- \frac{13}{u ^{2}} u^{2} : - 13

- \frac{13}{u ^{2}} u^{2}

Multiply fractions:

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

= - \frac{13 u ^{2}}{u ^{2}}

Cancel the common factor:

u^{2}

= - 13

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

- 3 u^{4} - 13 + 20 u^{2} = 0

- 3 u^{4} - 13 + 20 u^{2} = 0

- 3 u^{4} - 13 + 20 u^{2} = 0

Solve

- 3 u^{4} - 13 + 20 u^{2} = 0 : u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

- 3 u^{4} - 13 + 20 u^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 3 u^{4} + 20 u^{2} - 13 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

- 3 v^{2} + 20 v - 13 = 0

Solve

- 3 v^{2} + 20 v - 13 = 0 : v = \frac{10 - 61}{3}, v = \frac{10 + 61}{3}

- 3 v^{2} + 20 v - 13 = 0

Solve with the quadratic formula

- 3 v^{2} + 20 v - 13 = 0

Quadratic Equation Formula:

For

a = - 3, b = 20, c = - 13

v_{1, 2} = \frac{- 20 \pm 2 0 ^{2} - 4 ( - 3 ) ( - 13 )}{2 ( - 3 )}

v_{1, 2} = \frac{- 20 \pm 2 0 ^{2} - 4 ( - 3 ) ( - 13 )}{2 ( - 3 )}

2 0^{2} - 4 (- 3) (- 13) = 261

2 0^{2} - 4 (- 3) (- 13)

Apply rule

- (- a) = a

= 2 0^{2} - 4 \cdot 3 \cdot 13

Multiply the numbers:

4 \cdot 3 \cdot 13 = 156

= 2 0^{2} - 156

2 0^{2} = 400

= 400 - 156

Subtract the numbers:

400 - 156 = 244

= 244

Prime factorization of

244 : 2^{2} \cdot 61

244

244

divides by

2244 = 122 \cdot 2

= 2 \cdot 122

122

divides by

2122 = 61 \cdot 2

= 2 \cdot 2 \cdot 61

2, 61

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 61

= 2^{2} \cdot 61

= 2^{2} \cdot 61

Apply radical rule:

n ab = n a n b

= 61 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 261

v_{1, 2} = \frac{- 20 \pm 2 61}{2 ( - 3 )}

Separate the solutions

v_{1} = \frac{- 20 + 2 61}{2 ( - 3 )}, v_{2} = \frac{- 20 - 2 61}{2 ( - 3 )}

v = \frac{- 20 + 2 61}{2 ( - 3 )} : \frac{10 - 61}{3}

\frac{- 20 + 2 61}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 20 + 2 61}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 20 + 2 61}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 20 + 261 = - (20 - 261)

= \frac{20 - 2 61}{6}

Factor

20 - 261 : 2 (10 - 61)

20 - 261

Rewrite as

= 2 \cdot 10 - 261

Factor out common term

2

= 2 (10 - 61)

= \frac{2 ( 10 - 61 )}{6}

Cancel the common factor:

2

= \frac{10 - 61}{3}

v = \frac{- 20 - 2 61}{2 ( - 3 )} : \frac{10 + 61}{3}

\frac{- 20 - 2 61}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 20 - 2 61}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 20 - 2 61}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 20 - 261 = - (20 + 261)

= \frac{20 + 2 61}{6}

Factor

20 + 261 : 2 (10 + 61)

20 + 261

Rewrite as

= 2 \cdot 10 + 261

Factor out common term

2

= 2 (10 + 61)

= \frac{2 ( 10 + 61 )}{6}

Cancel the common factor:

2

= \frac{10 + 61}{3}

The solutions to the quadratic equation are:

v = \frac{10 - 61}{3}, v = \frac{10 + 61}{3}

v = \frac{10 - 61}{3}, v = \frac{10 + 61}{3}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = \frac{10 - 61}{3} : u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}

u^{2} = \frac{10 - 61}{3}

For

x^{2} = f (a)

the solutions are

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

u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}

Solve

u^{2} = \frac{10 + 61}{3} : u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

u^{2} = \frac{10 + 61}{3}

For

x^{2} = f (a)

the solutions are

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

u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

The solutions are

u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

5 (u^{2} - u^{- 2}) - 8 (u^{2} + u^{- 2}) + 20

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{10 - 61}{3} : x = \frac{1}{2} ln (\frac{10 - 61}{3})

e^{x} = \frac{10 - 61}{3}

Apply exponent rules

e^{x} = \frac{10 - 61}{3}

Apply exponent rule:

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

\frac{10 - 61}{3} = (\frac{10 - 61}{3})^{\frac{1}{2}}

e^{x} = (\frac{10 - 61}{3})^{\frac{1}{2}}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{10 - 61}{3})^{\frac{1}{2}}

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{10 - 61}{3})^{\frac{1}{2}}

Apply log rule:

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

ln (\frac{10 - 61}{3})^{\frac{1}{2}} = \frac{1}{2} ln (\frac{10 - 61}{3})

x = \frac{1}{2} ln (\frac{10 - 61}{3})

x = \frac{1}{2} ln (\frac{10 - 61}{3})

Solve

e^{x} = - \frac{10 - 61}{3} :

No Solution for

x \in R

e^{x} = - \frac{10 - 61}{3}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

Solve

e^{x} = \frac{10 + 61}{3} : x = \frac{1}{2} ln (\frac{10 + 61}{3})

e^{x} = \frac{10 + 61}{3}

Apply exponent rules

e^{x} = \frac{10 + 61}{3}

Apply exponent rule:

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

\frac{10 + 61}{3} = (\frac{10 + 61}{3})^{\frac{1}{2}}

e^{x} = (\frac{10 + 61}{3})^{\frac{1}{2}}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{10 + 61}{3})^{\frac{1}{2}}

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{10 + 61}{3})^{\frac{1}{2}}

Apply log rule:

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

ln (\frac{10 + 61}{3})^{\frac{1}{2}} = \frac{1}{2} ln (\frac{10 + 61}{3})

x = \frac{1}{2} ln (\frac{10 + 61}{3})

x = \frac{1}{2} ln (\frac{10 + 61}{3})

Solve

e^{x} = - \frac{10 + 61}{3} :

No Solution for

x \in R

e^{x} = - \frac{10 + 61}{3}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = \frac{1}{2} ln (\frac{10 - 61}{3}), x = \frac{1}{2} ln (\frac{10 + 61}{3})

x = \frac{1}{2} ln (\frac{10 - 61}{3}), x = \frac{1}{2} ln (\frac{10 + 61}{3})

Graph

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