What is the general solution for 4cosh(x)-sinh(x)=8 ?

The general solution for 4cosh(x)-sinh(x)=8 is x=ln(5),x=-ln(3)

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4cosh(x)-sinh(x)=8

Solution

4 cosh (x) - sinh (x) = 8

Solution

x = ln (5), x = - ln (3)

Degrees

x = 92.21399 \dots^{\circ}, x = - 62.94584 \dots^{\circ}

Solution steps

4 cosh (x) - sinh (x) = 8

Rewrite using trig identities

4 cosh (x) - sinh (x) = 8

Use the Hyperbolic identity:

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

4 cosh (x) - \frac{e ^{x} - e ^{- x}}{2} = 8

Use the Hyperbolic identity:

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

4 \cdot \frac{e ^{x} + e ^{- x}}{2} - \frac{e ^{x} - e ^{- x}}{2} = 8

4 \cdot \frac{e ^{x} + e ^{- x}}{2} - \frac{e ^{x} - e ^{- x}}{2} = 8

4 \cdot \frac{e ^{x} + e ^{- x}}{2} - \frac{e ^{x} - e ^{- x}}{2} = 8 : x = ln (5), x = - ln (3)

4 \cdot \frac{e ^{x} + e ^{- x}}{2} - \frac{e ^{x} - e ^{- x}}{2} = 8

Multiply both sides by

2

4 \cdot \frac{e ^{x} + e ^{- x}}{2} \cdot 2 - \frac{e ^{x} - e ^{- x}}{2} \cdot 2 = 8 \cdot 2

Simplify

4 (e^{x} + e^{- x}) - (e^{x} - e^{- x}) = 16

Apply exponent rules

4 (e^{x} + e^{- x}) - (e^{x} - e^{- x}) = 16

Apply exponent rule:

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

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

4 (e^{x} + (e^{x})^{- 1}) - (e^{x} - (e^{x})^{- 1}) = 16

4 (e^{x} + (e^{x})^{- 1}) - (e^{x} - (e^{x})^{- 1}) = 16

Rewrite the equation with

e^{x} = u

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

Solve

4 (u + u^{- 1}) - (u - u^{- 1}) = 16 : u = 5, u = \frac{1}{3}

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

Refine

4 (u + \frac{1}{u}) - (u - \frac{1}{u}) = 16

Simplify

- (u - \frac{1}{u}) : - u + \frac{1}{u}

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - u + \frac{1}{u}

4 (u + \frac{1}{u}) - u + \frac{1}{u} = 16

Multiply both sides by

u

4 (u + \frac{1}{u}) - u + \frac{1}{u} = 16

Multiply both sides by

u

4 (u + \frac{1}{u}) u - uu + \frac{1}{u} u = 16 u

Simplify

4 (u + \frac{1}{u}) u - uu + \frac{1}{u} u = 16 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

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

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

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

Expand

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

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

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

= 3 u^{2} + 4 + 1

Add the numbers:

4 + 1 = 5

= 3 u^{2} + 5

= 3 u^{2} + 5

3 u^{2} + 5 = 16 u

Solve

3 u^{2} + 5 = 16 u : u = 5, u = \frac{1}{3}

3 u^{2} + 5 = 16 u

Move

16 u

to the left side

3 u^{2} + 5 = 16 u

Subtract

16 u

from both sides

3 u^{2} + 5 - 16 u = 16 u - 16 u

Simplify

3 u^{2} + 5 - 16 u = 0

3 u^{2} + 5 - 16 u = 0

Write in the standard form

a x^{2} + b x + c = 0

3 u^{2} - 16 u + 5 = 0

Solve with the quadratic formula

3 u^{2} - 16 u + 5 = 0

Quadratic Equation Formula:

For

a = 3, b = - 16, c = 5

u_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 3 \cdot 5}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 3 \cdot 5}{2 \cdot 3}

(- 16)^{2} - 4 \cdot 3 \cdot 5 = 14

(- 16)^{2} - 4 \cdot 3 \cdot 5

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 16)^{2} = 1 6^{2}

= 1 6^{2} - 4 \cdot 3 \cdot 5

Multiply the numbers:

4 \cdot 3 \cdot 5 = 60

= 1 6^{2} - 60

1 6^{2} = 256

= 256 - 60

Subtract the numbers:

256 - 60 = 196

= 196

Factor the number:

196 = 1 4^{2}

= 1 4^{2}

Apply radical rule:

1 4^{2} = 14

= 14

u_{1, 2} = \frac{- ( - 16 ) \pm 14}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 16 ) + 14}{2 \cdot 3}, u_{2} = \frac{- ( - 16 ) - 14}{2 \cdot 3}

u = \frac{- ( - 16 ) + 14}{2 \cdot 3} : 5

\frac{- ( - 16 ) + 14}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{16 + 14}{2 \cdot 3}

Add the numbers:

16 + 14 = 30

= \frac{30}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{30}{6}

Divide the numbers:

\frac{30}{6} = 5

= 5

u = \frac{- ( - 16 ) - 14}{2 \cdot 3} : \frac{1}{3}

\frac{- ( - 16 ) - 14}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{16 - 14}{2 \cdot 3}

Subtract the numbers:

16 - 14 = 2

= \frac{2}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2}{6}

Cancel the common factor:

2

= \frac{1}{3}

The solutions to the quadratic equation are:

u = 5, u = \frac{1}{3}

u = 5, u = \frac{1}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 5, u = \frac{1}{3}

u = 5, u = \frac{1}{3}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 5 : x = ln (5)

e^{x} = 5

Apply exponent rules

e^{x} = 5

f (x) = g (x)

, then

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

ln (e^{x}) = ln (5)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (5)

x = ln (5)

Solve

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

e^{x} = \frac{1}{3}

Apply exponent rules

e^{x} = \frac{1}{3}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{1}{3})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{1}{3})

Simplify

ln (\frac{1}{3}) : - ln (3)

ln (\frac{1}{3})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (3)

x = - ln (3)

x = - ln (3)

x = ln (5), x = - ln (3)

x = ln (5), x = - ln (3)

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Frequently Asked Questions (FAQ)

What is the general solution for 4cosh(x)-sinh(x)=8 ?
The general solution for 4cosh(x)-sinh(x)=8 is x=ln(5),x=-ln(3)