What is the general solution for tanh^2(x)+5sech(x)-5=0 ?

The general solution for tanh^2(x)+5sech(x)-5=0 is x=0

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tanh^2(x)+5sech(x)-5=0

Solution

tanh^{2} (x) + 5 sech (x) - 5 = 0

Solution

x = 0

Degrees

x = 0^{\circ}

Solution steps

tanh^{2} (x) + 5 sech (x) - 5 = 0

Rewrite using trig identities

tanh^{2} (x) + 5 sech (x) - 5 = 0

Use the Hyperbolic identity:

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

(\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} + 5 sech (x) - 5 = 0

Use the Hyperbolic identity:

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

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

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

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

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

Apply exponent rules

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

Apply exponent rule:

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

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

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

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

Rewrite the equation with

e^{x} = u

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

Solve

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

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

Refine

\frac{( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} + \frac{10 u}{u ^{2} + 1} - 5 = 0

Multiply by LCM

\frac{( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} + \frac{10 u}{u ^{2} + 1} - 5 = 0

Find Least Common Multiplier of

(u^{2} + 1)^{2}, u^{2} + 1 : (u^{2} + 1)^{2}

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

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

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

u^{2} + 1

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

Multiply by LCM=

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

\frac{( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} (u^{2} + 1)^{2} + \frac{10 u}{u ^{2} + 1} (u^{2} + 1)^{2} - 5 (u^{2} + 1)^{2} = 0 \cdot (u^{2} + 1)^{2}

Simplify

\frac{( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} (u^{2} + 1)^{2} + \frac{10 u}{u ^{2} + 1} (u^{2} + 1)^{2} - 5 (u^{2} + 1)^{2} = 0 \cdot (u^{2} + 1)^{2}

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

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

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

Simplify

\frac{10 u}{u ^{2} + 1} (u^{2} + 1)^{2} : 10 u (u^{2} + 1)

\frac{10 u}{u ^{2} + 1} (u^{2} + 1)^{2}

Multiply fractions:

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

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

Cancel the common factor:

u^{2} + 1

= 10 u (u^{2} + 1)

Simplify

0 \cdot (u^{2} + 1)^{2} : 0

0 \cdot (u^{2} + 1)^{2}

Apply rule

0 \cdot a = 0

= 0

(u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2} = 0

(u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2} = 0

(u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2} = 0

Solve

(u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2} = 0 : u = 1

(u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2} = 0

Factor

(u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2} : - 2 (u - 1)^{2} (2 u^{2} - u + 2)

(u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2}

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

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

Factor

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

Factor

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

u^{2} - 1

Rewrite

1

1^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (u + 1) (u - 1)

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

Apply exponent rule:

(ab)^{n} = a^{n} b^{n}

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

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

= (u + 1)^{2} (u - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2}

Expand

(u + 1)^{2} (u - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2} : - 4 u^{4} + 10 u^{3} - 12 u^{2} + 10 u - 4

(u + 1)^{2} (u - 1)^{2} + 10 u (u^{2} + 1) - 5 (u^{2} + 1)^{2}

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

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

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

(u + 1)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = u, b = 1

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

Multiply the numbers:

2 \cdot 1 = 2

= u^{2} + 2 u + 1

= u^{2} + 2 u + 1

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

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

(u - 1)^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = u, b = 1

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

Simplify

u^{2} - 2 u \cdot 1 + 1^{2} : u^{2} - 2 u + 1

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

Apply rule

1^{a} = 1

1^{2} = 1

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

Multiply the numbers:

2 \cdot 1 = 2

= u^{2} - 2 u + 1

= u^{2} - 2 u + 1

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

5 (u^{2} + 1)^{2} = 5 (u^{4} + 2 u^{2} + 1)

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

(u^{2} + 1)^{2} = u^{4} + 2 u^{2} + 1

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

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

Simplify

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2} : u^{4} + 2 u^{2} + 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} + 2 \cdot 1 \cdot u^{2} + 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Apply exponent rule:

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

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

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

2 u^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

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

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

= 5 (u^{4} + 2 u^{2} + 1)

= (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) + 10 u (u^{2} + 1) - 5 (u^{4} + 2 u^{2} + 1)

Expand

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

= u^{2} u^{2} - 2 u^{2} u + 1 \cdot u^{2} + 2 u^{2} u - 2 \cdot 2 uu + 2 \cdot 1 \cdot u + 1 \cdot u^{2} - 1 \cdot 2 u + 1 \cdot 1

Simplify

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

u^{2} u^{2} - 2 u^{2} u + 1 \cdot u^{2} + 2 u^{2} u - 2 \cdot 2 uu + 2 \cdot 1 \cdot u + 1 \cdot u^{2} - 1 \cdot 2 u + 1 \cdot 1

Group like terms

= u^{2} u^{2} - 2 u^{2} u + 1 \cdot u^{2} + 2 u^{2} u + 1 \cdot u^{2} - 2 \cdot 2 uu + 2 \cdot 1 \cdot u - 1 \cdot 2 u + 1 \cdot 1

Add similar elements:

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

= u^{2} u^{2} - 2 u^{2} u + 2 u^{2} + 2 u^{2} u - 2 \cdot 2 uu + 2 \cdot 1 \cdot u - 1 \cdot 2 u + 1 \cdot 1

Add similar elements:

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

= u^{2} u^{2} + 2 u^{2} - 2 \cdot 2 uu + 2 \cdot 1 \cdot u - 1 \cdot 2 u + 1 \cdot 1

Add similar elements:

2 \cdot 1 \cdot u - 1 \cdot 2 u = 0

= u^{2} u^{2} + 2 u^{2} - 2 \cdot 2 uu + 1 \cdot 1

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}

2 \cdot 2 uu = 4 u^{2}

2 \cdot 2 uu

Multiply the numbers:

2 \cdot 2 = 4

= 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}

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

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

Add similar elements:

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

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

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

= u^{4} - 2 u^{2} + 1 + 10 u (u^{2} + 1) - 5 (u^{4} + 2 u^{2} + 1)

Expand

10 u (u^{2} + 1) : 10 u^{3} + 10 u

10 u (u^{2} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 10 u, b = u^{2}, c = 1

= 10 u u^{2} + 10 u \cdot 1

= 10 u^{2} u + 10 \cdot 1 \cdot u

Simplify

10 u^{2} u + 10 \cdot 1 \cdot u : 10 u^{3} + 10 u

10 u^{2} u + 10 \cdot 1 \cdot u

10 u^{2} u = 10 u^{3}

10 u^{2} u

Apply exponent rule:

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

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

= 10 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 10 u^{3}

10 \cdot 1 \cdot u = 10 u

10 \cdot 1 \cdot u

Multiply the numbers:

10 \cdot 1 = 10

= 10 u

= 10 u^{3} + 10 u

= 10 u^{3} + 10 u

= u^{4} - 2 u^{2} + 1 + 10 u^{3} + 10 u - 5 (u^{4} + 2 u^{2} + 1)

Expand

- 5 (u^{4} + 2 u^{2} + 1) : - 5 u^{4} - 10 u^{2} - 5

- 5 (u^{4} + 2 u^{2} + 1)

Distribute parentheses

= (- 5) u^{4} + (- 5) \cdot 2 u^{2} + (- 5) \cdot 1

Apply minus-plus rules

+ (- a) = - a

= - 5 u^{4} - 5 \cdot 2 u^{2} - 5 \cdot 1

Simplify

- 5 u^{4} - 5 \cdot 2 u^{2} - 5 \cdot 1 : - 5 u^{4} - 10 u^{2} - 5

- 5 u^{4} - 5 \cdot 2 u^{2} - 5 \cdot 1

Multiply the numbers:

5 \cdot 2 = 10

= - 5 u^{4} - 10 u^{2} - 5 \cdot 1

Multiply the numbers:

5 \cdot 1 = 5

= - 5 u^{4} - 10 u^{2} - 5

= - 5 u^{4} - 10 u^{2} - 5

= u^{4} - 2 u^{2} + 1 + 10 u^{3} + 10 u - 5 u^{4} - 10 u^{2} - 5

Simplify

u^{4} - 2 u^{2} + 1 + 10 u^{3} + 10 u - 5 u^{4} - 10 u^{2} - 5 : - 4 u^{4} + 10 u^{3} - 12 u^{2} + 10 u - 4

u^{4} - 2 u^{2} + 1 + 10 u^{3} + 10 u - 5 u^{4} - 10 u^{2} - 5

Group like terms

= u^{4} - 5 u^{4} + 10 u^{3} - 2 u^{2} - 10 u^{2} + 10 u + 1 - 5

Add similar elements:

- 2 u^{2} - 10 u^{2} = - 12 u^{2}

= u^{4} - 5 u^{4} + 10 u^{3} - 12 u^{2} + 10 u + 1 - 5

Add similar elements:

u^{4} - 5 u^{4} = - 4 u^{4}

= - 4 u^{4} + 10 u^{3} - 12 u^{2} + 10 u + 1 - 5

Add/Subtract the numbers:

1 - 5 = - 4

= - 4 u^{4} + 10 u^{3} - 12 u^{2} + 10 u - 4

= - 4 u^{4} + 10 u^{3} - 12 u^{2} + 10 u - 4

= - 4 u^{4} + 10 u^{3} - 12 u^{2} + 10 u - 4

Factor

- 4 u^{4} + 10 u^{3} - 12 u^{2} + 10 u - 4 : - 2 (u - 1)^{2} (2 u^{2} - u + 2)

- 4 u^{4} + 10 u^{3} - 12 u^{2} + 10 u - 4

Factor out common term

- 2 : - 2 (2 u^{4} - 5 u^{3} + 6 u^{2} - 5 u + 2)

- 4 u^{4} + 10 u^{3} - 12 u^{2} + 10 u - 4

Rewrite

4

2 \cdot 2

Rewrite

10

2 \cdot 5

= - 2 \cdot 2 u^{2 \cdot 2} + 2 \cdot 5 u^{3} - 2 \cdot 6 u^{2} + 2 \cdot 5 u - 2 \cdot 2

Factor out common term

- 2

= - 2 (2 u^{4} - 5 u^{3} + 6 u^{2} - 5 u + 2)

= - 2 (2 u^{4} - 5 u^{3} + 6 u^{2} - 5 u + 2)

Factor

2 u^{4} - 5 u^{3} + 6 u^{2} - 5 u + 2 : (u - 1) (u - 1) (2 u^{2} - u + 2)

2 u^{4} - 5 u^{3} + 6 u^{2} - 5 u + 2

Use the rational root theorem

a_{0} = 2, a_{n} = 2

The dividers of

a_{0} : 1, 2,

The dividers of

a_{n} : 1, 2

Therefore, check the following rational numbers:

\pm \frac{1 , 2}{1 , 2}

\frac{1}{1}

is a root of the expression, so factor out

u - 1

= (u - 1) \frac{2 u ^{4} - 5 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1}

\frac{2 u ^{4} - 5 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1} = 2 u^{3} - 3 u^{2} + 3 u - 2

\frac{2 u ^{4} - 5 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1}

Divide

\frac{2 u ^{4} - 5 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1} : \frac{2 u ^{4} - 5 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1} = 2 u^{3} + \frac{- 3 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1}

Divide the leading coefficients of the numerator

2 u^{4} - 5 u^{3} + 6 u^{2} - 5 u + 2

and the divisor

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

Quotient = 2 u^{3}

Multiply

u - 1

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

Subtract

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

from

2 u^{4} - 5 u^{3} + 6 u^{2} - 5 u + 2

to get new remainder

Remainder = - 3 u^{3} + 6 u^{2} - 5 u + 2

Therefore

\frac{2 u ^{4} - 5 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1} = 2 u^{3} + \frac{- 3 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1}

= 2 u^{3} + \frac{- 3 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1}

Divide

\frac{- 3 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1} : \frac{- 3 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1} = - 3 u^{2} + \frac{3 u ^{2} - 5 u + 2}{u - 1}

Divide the leading coefficients of the numerator

- 3 u^{3} + 6 u^{2} - 5 u + 2

and the divisor

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

Quotient = - 3 u^{2}

Multiply

u - 1

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

Subtract

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

from

- 3 u^{3} + 6 u^{2} - 5 u + 2

to get new remainder

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

Therefore

\frac{- 3 u ^{3} + 6 u ^{2} - 5 u + 2}{u - 1} = - 3 u^{2} + \frac{3 u ^{2} - 5 u + 2}{u - 1}

= 2 u^{3} - 3 u^{2} + \frac{3 u ^{2} - 5 u + 2}{u - 1}

Divide

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

Divide the leading coefficients of the numerator

3 u^{2} - 5 u + 2

and the divisor

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

Quotient = 3 u

Multiply

u - 1

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

Subtract

3 u^{2} - 3 u

from

3 u^{2} - 5 u + 2

to get new remainder

Remainder = - 2 u + 2

Therefore

\frac{3 u ^{2} - 5 u + 2}{u - 1} = 3 u + \frac{- 2 u + 2}{u - 1}

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

Divide

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

Divide the leading coefficients of the numerator

- 2 u + 2

and the divisor

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

Quotient = - 2

Multiply

u - 1

- 2 : - 2 u + 2

Subtract

- 2 u + 2

from

- 2 u + 2

to get new remainder

Remainder = 0

Therefore

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

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

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

Factor

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

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

Use the rational root theorem

a_{0} = 2, a_{n} = 2

The dividers of

a_{0} : 1, 2,

The dividers of

a_{n} : 1, 2

Therefore, check the following rational numbers:

\pm \frac{1 , 2}{1 , 2}

\frac{1}{1}

is a root of the expression, so factor out

u - 1

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

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

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

Divide

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

Divide the leading coefficients of the numerator

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

and the divisor

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

Quotient = 2 u^{2}

Multiply

u - 1

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

Subtract

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

from

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

to get new remainder

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

Therefore

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

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

Divide

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

Divide the leading coefficients of the numerator

- u^{2} + 3 u - 2

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} + 3 u - 2

to get new remainder

Remainder = 2 u - 2

Therefore

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

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

Divide

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

Divide the leading coefficients of the numerator

2 u - 2

and the divisor

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

Quotient = 2

Multiply

u - 1

2 : 2 u - 2

Subtract

2 u - 2

from

2 u - 2

to get new remainder

Remainder = 0

Therefore

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

= 2 u^{2} - u + 2

= 2 u^{2} - u + 2

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

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

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

Refine

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

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

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

No Solution for

u \in R

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

Discriminant

2 u^{2} - u + 2 = 0 : - 15

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

For a quadratic equation of the form

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

the discriminant is

b^{2} - 4 a c

For

a = 2, b = - 1, c = 2 : (- 1)^{2} - 4 \cdot 2 \cdot 2

(- 1)^{2} - 4 \cdot 2 \cdot 2

Expand

(- 1)^{2} - 4 \cdot 2 \cdot 2 : - 15

(- 1)^{2} - 4 \cdot 2 \cdot 2

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 2 \cdot 2 = 16

4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 16

= 1 - 16

Subtract the numbers:

1 - 16 = - 15

= - 15

- 15

Discriminant cannot be negative for

u \in R

The solution is

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

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

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)

What is the general solution for tanh^2(x)+5sech(x)-5=0 ?
The general solution for tanh^2(x)+5sech(x)-5=0 is x=0