What is the general solution for solvefor x,cosh(sqrt(x))=0 ?

The general solution for solvefor x,cosh(sqrt(x))=0 is No Solution for x\in\mathbb{R}

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solvefor x,cosh(sqrt(x))=0

Solution

solve for

x, cosh (x) = 0

Solution

No Solution for x \in R

Solution steps

cosh (x) = 0

Rewrite using trig identities

cosh (x) = 0

Use the Hyperbolic identity:

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

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

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

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

No Solution for

x \in R

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

e^{x} + e^{- x} = 0

Subtract

e^{- x}

from both sides

e^{x} + e^{- x} - e^{- x} = 0 - e^{- x}

Simplify

e^{x} = - e^{- x}

Apply exponent rules

e^{x} = - e^{- x}

f (x) = g (x)

, then

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

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

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

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

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

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

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

Apply log rule:

ln (e^{a}) = a

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

x = ln (- 1) - x

x = ln (- 1) - x

Solve

x = ln (- 1) - x :

No Solution for

x \in R

x = ln (- 1) - x

Expand

ln (- 1) - x :

Undefined

ln (- 1) - x

lo g_{a} (b), b < 0

is undefined

= Undefined

x = Undefined

Square both sides:

x =

Undefined

^{2}

x = Undefined

(x)^{2} = Undefined^{2}

Expand

(x)^{2} : x

(x)^{2}

Apply radical rule:

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

= (x^{\frac{1}{2}})^{2}

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= x

x = Undefined^{2}

x = Undefined^{2}

Verify Solutions:

x =

Undefined

^{2}

False

Check the solutions by plugging them into

x = ln (- 1) - x

Remove the ones that don't agree with the equation.

Plug in

x =

Undefined

^{2} :

False

Undefined^{2} = ln (- 1) - Undefined^{2}

Simplify

ln (- 1) - Undefined^{2} :

Undefined

ln (- 1) - Undefined^{2}

lo g_{a} (b), b < 0

is undefined

= Undefined

Undefined^{2} =

Undefined

False

The solution is

No Solution for x \in R

No Solution for x \in R

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,cosh(sqrt(x))=0 ?
The general solution for solvefor x,cosh(sqrt(x))=0 is No Solution for x\in\mathbb{R}