What is the general solution for cosh(2x)=2cosh(x)-1 ?

The general solution for cosh(2x)=2cosh(x)-1 is x=0

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cosh(2x)=2cosh(x)-1

Solution

cosh (2 x) = 2 cosh (x) - 1

Solution

x = 0

Degrees

x = 0^{\circ}

Solution steps

cosh (2 x) = 2 cosh (x) - 1

Rewrite using trig identities

cosh (2 x) = 2 cosh (x) - 1

Use the Hyperbolic identity:

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

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

Use the Hyperbolic identity:

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

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

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

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

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

Multiply both sides by

2

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

Simplify

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

Apply exponent rules

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

Apply exponent rule:

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

e^{2 x} = (e^{x})^{2}, e^{- 2 x} = (e^{x})^{- 2}, e^{- x} = (e^{x})^{- 1}

(e^{x})^{2} + (e^{x})^{- 2} = 2 (e^{x} + (e^{x})^{- 1}) - 2

(e^{x})^{2} + (e^{x})^{- 2} = 2 (e^{x} + (e^{x})^{- 1}) - 2

Rewrite the equation with

e^{x} = u

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

Solve

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

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

Refine

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

Multiply both sides by

u^{2}

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

Multiply both sides by

u^{2}

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

Simplify

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

Simplify

u^{2} u^{2} : u^{4}

u^{2} u^{2}

Apply exponent rule:

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

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

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

Simplify

\frac{1}{u ^{2}} u^{2} : 1

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

Multiply fractions:

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

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

Cancel the common factor:

u^{2}

= 1

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

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

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

Expand

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

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

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

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

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

= 2 u^{2} u + 2 u^{2} \frac{1}{u}

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

Simplify

2 u^{2} u + 2 \cdot \frac{1}{u} u^{2} : 2 u^{3} + 2 u

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

2 u^{2} u = 2 u^{3}

2 u^{2} u

Apply exponent rule:

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

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

= 2 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2 u^{3}

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

Cancel the common factor:

u

= 2 u

= 2 u^{3} + 2 u

= 2 u^{3} + 2 u

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

u^{4} + 1 = 2 u^{3} + 2 u - 2 u^{2}

Solve

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

u^{4} + 1 = 2 u^{3} + 2 u - 2 u^{2}

Move

2 u^{2}

to the left side

u^{4} + 1 = 2 u^{3} + 2 u - 2 u^{2}

Add

2 u^{2}

to both sides

u^{4} + 1 + 2 u^{2} = 2 u^{3} + 2 u - 2 u^{2} + 2 u^{2}

Simplify

u^{4} + 1 + 2 u^{2} = 2 u^{3} + 2 u

u^{4} + 1 + 2 u^{2} = 2 u^{3} + 2 u

Move

2 u

to the left side

u^{4} + 1 + 2 u^{2} = 2 u^{3} + 2 u

Subtract

2 u

from both sides

u^{4} + 1 + 2 u^{2} - 2 u = 2 u^{3} + 2 u - 2 u

Simplify

u^{4} + 1 + 2 u^{2} - 2 u = 2 u^{3}

u^{4} + 1 + 2 u^{2} - 2 u = 2 u^{3}

Move

2 u^{3}

to the left side

u^{4} + 1 + 2 u^{2} - 2 u = 2 u^{3}

Subtract

2 u^{3}

from both sides

u^{4} + 1 + 2 u^{2} - 2 u - 2 u^{3} = 2 u^{3} - 2 u^{3}

Simplify

u^{4} + 1 + 2 u^{2} - 2 u - 2 u^{3} = 0

u^{4} + 1 + 2 u^{2} - 2 u - 2 u^{3} = 0

Write in the standard form

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

u^{4} - 2 u^{3} + 2 u^{2} - 2 u + 1 = 0

Factor

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

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

Use the rational root theorem

a_{0} = 1, a_{n} = 1

The dividers of

a_{0} : 1,

The dividers of

a_{n} : 1

Therefore, check the following rational numbers:

\pm \frac{1}{1}

\frac{1}{1}

is a root of the expression, so factor out

u - 1

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

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

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

Divide

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

Divide the leading coefficients of the numerator

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

and the divisor

u - 1 : \frac{u ^{4}}{u} = u^{3}

Quotient = u^{3}

Multiply

u - 1

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

Subtract

u^{4} - u^{3}

from

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

to get new remainder

Remainder = - u^{3} + 2 u^{2} - 2 u + 1

Therefore

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

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

Divide

\frac{- u ^{3} + 2 u ^{2} - 2 u + 1}{u - 1} : \frac{- u ^{3} + 2 u ^{2} - 2 u + 1}{u - 1} = - u^{2} + \frac{u ^{2} - 2 u + 1}{u - 1}

Divide the leading coefficients of the numerator

- u^{3} + 2 u^{2} - 2 u + 1

and the divisor

u - 1 : \frac{- u ^{3}}{u} = - u^{2}

Quotient = - u^{2}

Multiply

u - 1

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

Subtract

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

from

- u^{3} + 2 u^{2} - 2 u + 1

to get new remainder

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

Therefore

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

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

Divide

\frac{u ^{2} - 2 u + 1}{u - 1} : \frac{u ^{2} - 2 u + 1}{u - 1} = u + \frac{- u + 1}{u - 1}

Divide the leading coefficients of the numerator

u^{2} - 2 u + 1

and the divisor

u - 1 : \frac{u ^{2}}{u} = u

Quotient = u

Multiply

u - 1

u : u^{2} - u

Subtract

u^{2} - u

from

u^{2} - 2 u + 1

to get new remainder

Remainder = - u + 1

Therefore

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

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

Divide

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

Divide the leading coefficients of the numerator

- u + 1

and the divisor

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

Quotient = - 1

Multiply

u - 1

- 1 : - u + 1

Subtract

- u + 1

from

- u + 1

to get new remainder

Remainder = 0

Therefore

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

= u^{3} - u^{2} + u - 1

= u^{3} - u^{2} + u - 1

Factor

u^{3} - u^{2} + u - 1 : (u - 1) (u^{2} + 1)

u^{3} - u^{2} + u - 1

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

Factor out

u^{2}

from

u^{3} - u^{2} : u^{2} (u - 1)

u^{3} - u^{2}

Apply exponent rule:

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

u^{3} = u u^{2}

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

Factor out common term

u^{2}

= u^{2} (u - 1)

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

Factor out common term

u - 1

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

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

Refine

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u - 1 = 0 or u^{2} + 1 = 0

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

Solve

u^{2} + 1 = 0 :

No Solution for

u \in R

u^{2} + 1 = 0

Move

1

to the right side

u^{2} + 1 = 0

Subtract

1

from both sides

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

Simplify

u^{2} = - 1

u^{2} = - 1

x^{2}

cannot be negative for

x \in R

No Solution for u \in R

The solution is

u = 1

u = 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u^{2} + u^{- 2}

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

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

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1

u = 1

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 1 : x = 0

e^{x} = 1

Apply exponent rules

e^{x} = 1

f (x) = g (x)

, then

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

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

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (1)

Simplify

ln (1) : 0

ln (1)

Apply log rule:

lo g_{a} (1) = 0

= 0

x = 0

x = 0

x = 0

x = 0

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Frequently Asked Questions (FAQ)

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