What is the general solution for cosh(x)+4sinh(x)+2=0 ?

The general solution for cosh(x)+4sinh(x)+2=0 is x=ln((-2+sqrt(19))/5)

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cosh(x)+4sinh(x)+2=0

Solution

cosh (x) + 4 sinh (x) + 2 = 0

Solution

x = ln (\frac{- 2 + 19}{5})

Degrees

x = - 43.04305 \dots^{\circ}

Solution steps

cosh (x) + 4 sinh (x) + 2 = 0

Rewrite using trig identities

cosh (x) + 4 sinh (x) + 2 = 0

Use the Hyperbolic identity:

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

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

Use the Hyperbolic identity:

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

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

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

\frac{e ^{x} + e ^{- x}}{2} + 4 \cdot \frac{e ^{x} - e ^{- x}}{2} + 2 = 0 : x = ln (\frac{- 2 + 19}{5})

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

Multiply both sides by

2

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

Simplify

e^{x} + e^{- x} + 4 (e^{x} - e^{- x}) + 4 = 0

Apply exponent rules

e^{x} + e^{- x} + 4 (e^{x} - e^{- x}) + 4 = 0

Apply exponent rule:

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

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

e^{x} + (e^{x})^{- 1} + 4 (e^{x} - (e^{x})^{- 1}) + 4 = 0

e^{x} + (e^{x})^{- 1} + 4 (e^{x} - (e^{x})^{- 1}) + 4 = 0

Rewrite the equation with

e^{x} = u

u + (u)^{- 1} + 4 (u - (u)^{- 1}) + 4 = 0

Solve

u + u^{- 1} + 4 (u - u^{- 1}) + 4 = 0 : u = \frac{- 2 + 19}{5}, u = - \frac{2 + 19}{5}

u + u^{- 1} + 4 (u - u^{- 1}) + 4 = 0

Refine

u + \frac{1}{u} + 4 (u - \frac{1}{u}) + 4 = 0

Multiply both sides by

u

u + \frac{1}{u} + 4 (u - \frac{1}{u}) + 4 = 0

Multiply both sides by

u

uu + \frac{1}{u} u + 4 (u - \frac{1}{u}) u + 4 u = 0 \cdot u

Simplify

uu + \frac{1}{u} u + 4 (u - \frac{1}{u}) u + 4 u = 0 \cdot u

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

\frac{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

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

= \frac{1 \cdot u}{u}

Cancel the common factor:

u

= 1

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} + 1 + 4 (u - \frac{1}{u}) u + 4 u = 0

u^{2} + 1 + 4 (u - \frac{1}{u}) u + 4 u = 0

u^{2} + 1 + 4 (u - \frac{1}{u}) u + 4 u = 0

Expand

u^{2} + 1 + 4 (u - \frac{1}{u}) u + 4 u : 5 u^{2} + 4 u - 3

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

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

Expand

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

4 u (u - \frac{1}{u})

Apply the distributive law:

a (b - c) = ab - a c

a = 4 u, b = u, c = \frac{1}{u}

= 4 uu - 4 u \frac{1}{u}

= 4 uu - 4 \cdot \frac{1}{u} u

Simplify

4 uu - 4 \cdot \frac{1}{u} u : 4 u^{2} - 4

4 uu - 4 \cdot \frac{1}{u} u

4 uu = 4 u^{2}

4 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 4 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 4 u^{2}

4 \cdot \frac{1}{u} u = 4

4 \cdot \frac{1}{u} u

Multiply fractions:

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

= \frac{1 \cdot 4 u}{u}

Cancel the common factor:

u

= 1 \cdot 4

Multiply the numbers:

1 \cdot 4 = 4

= 4

= 4 u^{2} - 4

= 4 u^{2} - 4

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

Simplify

u^{2} + 1 + 4 u^{2} - 4 + 4 u : 5 u^{2} + 4 u - 3

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

Group like terms

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

Add similar elements:

u^{2} + 4 u^{2} = 5 u^{2}

= 5 u^{2} + 4 u + 1 - 4

Add/Subtract the numbers:

1 - 4 = - 3

= 5 u^{2} + 4 u - 3

= 5 u^{2} + 4 u - 3

5 u^{2} + 4 u - 3 = 0

Solve

5 u^{2} + 4 u - 3 = 0 : u = \frac{- 2 + 19}{5}, u = - \frac{2 + 19}{5}

5 u^{2} + 4 u - 3 = 0

Solve with the quadratic formula

5 u^{2} + 4 u - 3 = 0

Quadratic Equation Formula:

For

a = 5, b = 4, c = - 3

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

4^{2} - 4 \cdot 5 (- 3) = 219

4^{2} - 4 \cdot 5 (- 3)

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 5 \cdot 3

Multiply the numbers:

4 \cdot 5 \cdot 3 = 60

= 4^{2} + 60

4^{2} = 16

= 16 + 60

Add the numbers:

16 + 60 = 76

= 76

Prime factorization of

76 : 2^{2} \cdot 19

76

76

divides by

276 = 38 \cdot 2

= 2 \cdot 38

38

divides by

238 = 19 \cdot 2

= 2 \cdot 2 \cdot 19

2, 19

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 19

= 2^{2} \cdot 19

= 2^{2} \cdot 19

Apply radical rule:

= 19 2^{2}

Apply radical rule:

2^{2} = 2

= 219

u_{1, 2} = \frac{- 4 \pm 2 19}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- 4 + 2 19}{2 \cdot 5}, u_{2} = \frac{- 4 - 2 19}{2 \cdot 5}

u = \frac{- 4 + 2 19}{2 \cdot 5} : \frac{- 2 + 19}{5}

\frac{- 4 + 2 19}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 4 + 2 19}{10}

Factor

- 4 + 219 : 2 (- 2 + 19)

- 4 + 219

Rewrite as

= - 2 \cdot 2 + 219

Factor out common term

2

= 2 (- 2 + 19)

= \frac{2 ( - 2 + 19 )}{10}

Cancel the common factor:

2

= \frac{- 2 + 19}{5}

u = \frac{- 4 - 2 19}{2 \cdot 5} : - \frac{2 + 19}{5}

\frac{- 4 - 2 19}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 4 - 2 19}{10}

Factor

- 4 - 219 : - 2 (2 + 19)

- 4 - 219

Rewrite as

= - 2 \cdot 2 - 219

Factor out common term

2

= - 2 (2 + 19)

= - \frac{2 ( 2 + 19 )}{10}

Cancel the common factor:

2

= - \frac{2 + 19}{5}

The solutions to the quadratic equation are:

u = \frac{- 2 + 19}{5}, u = - \frac{2 + 19}{5}

u = \frac{- 2 + 19}{5}, u = - \frac{2 + 19}{5}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{- 2 + 19}{5}, u = - \frac{2 + 19}{5}

u = \frac{- 2 + 19}{5}, u = - \frac{2 + 19}{5}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{- 2 + 19}{5} : x = ln (\frac{- 2 + 19}{5})

e^{x} = \frac{- 2 + 19}{5}

Apply exponent rules

e^{x} = \frac{- 2 + 19}{5}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{- 2 + 19}{5})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{- 2 + 19}{5})

x = ln (\frac{- 2 + 19}{5})

Solve

e^{x} = - \frac{2 + 19}{5} :

No Solution for

x \in R

e^{x} = - \frac{2 + 19}{5}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (\frac{- 2 + 19}{5})

x = ln (\frac{- 2 + 19}{5})

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Frequently Asked Questions (FAQ)

What is the general solution for cosh(x)+4sinh(x)+2=0 ?
The general solution for cosh(x)+4sinh(x)+2=0 is x=ln((-2+sqrt(19))/5)