What is the general solution for 12cosh(2x)+7sinh(x)-24=0 ?

The general solution for 12cosh(2x)+7sinh(x)-24=0 is x=ln(1.73025…),x=ln(0.45623…)

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Popular Trigonometry >

12cosh(2x)+7sinh(x)-24=0

Solution

12 cosh (2 x) + 7 sinh (x) - 24 = 0

Solution

x = ln (1.73025 \dots), x = ln (0.45623 \dots)

Degrees

x = 31.41360 \dots^{\circ}, x = - 44.96326 \dots^{\circ}

Solution steps

12 cosh (2 x) + 7 sinh (x) - 24 = 0

Rewrite using trig identities

12 cosh (2 x) + 7 sinh (x) - 24 = 0

Use the Hyperbolic identity:

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

12 cosh (2 x) + 7 \cdot \frac{e ^{x} - e ^{- x}}{2} - 24 = 0

Use the Hyperbolic identity:

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

12 \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 7 \cdot \frac{e ^{x} - e ^{- x}}{2} - 24 = 0

12 \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 7 \cdot \frac{e ^{x} - e ^{- x}}{2} - 24 = 0

12 \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 7 \cdot \frac{e ^{x} - e ^{- x}}{2} - 24 = 0 : x = ln (1.73025 \dots), x = ln (0.45623 \dots)

12 \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 7 \cdot \frac{e ^{x} - e ^{- x}}{2} - 24 = 0

Apply exponent rules

12 \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 7 \cdot \frac{e ^{x} - e ^{- x}}{2} - 24 = 0

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}

12 \cdot \frac{( e ^{x} ) ^{2} + ( e ^{x} ) ^{- 2}}{2} + 7 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} - 24 = 0

12 \cdot \frac{( e ^{x} ) ^{2} + ( e ^{x} ) ^{- 2}}{2} + 7 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} - 24 = 0

Rewrite the equation with

e^{x} = u

12 \cdot \frac{( u ) ^{2} + ( u ) ^{- 2}}{2} + 7 \cdot \frac{u - ( u ) ^{- 1}}{2} - 24 = 0

Solve

12 \cdot \frac{u ^{2} + u ^{- 2}}{2} + 7 \cdot \frac{u - u ^{- 1}}{2} - 24 = 0 : u \approx 1.73025 \dots, u \approx 0.45623 \dots, u \approx - 0.57794 \dots, u \approx - 2.19187 \dots

12 \cdot \frac{u ^{2} + u ^{- 2}}{2} + 7 \cdot \frac{u - u ^{- 1}}{2} - 24 = 0

Refine

\frac{6 ( u ^{4} + 1 )}{u ^{2}} + \frac{7 ( u ^{2} - 1 )}{2 u} - 24 = 0

Multiply by LCM

\frac{6 ( u ^{4} + 1 )}{u ^{2}} + \frac{7 ( u ^{2} - 1 )}{2 u} - 24 = 0

Find Least Common Multiplier of

u^{2}, 2 u : 2 u^{2}

u^{2}, 2 u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u^{2}

2 u

= 2 u^{2}

Multiply by LCM=

2 u^{2}

\frac{6 ( u ^{4} + 1 )}{u ^{2}} \cdot 2 u^{2} + \frac{7 ( u ^{2} - 1 )}{2 u} \cdot 2 u^{2} - 24 \cdot 2 u^{2} = 0 \cdot 2 u^{2}

Simplify

\frac{6 ( u ^{4} + 1 )}{u ^{2}} \cdot 2 u^{2} + \frac{7 ( u ^{2} - 1 )}{2 u} \cdot 2 u^{2} - 24 \cdot 2 u^{2} = 0 \cdot 2 u^{2}

Simplify

\frac{6 ( u ^{4} + 1 )}{u ^{2}} \cdot 2 u^{2} : 12 (u^{4} + 1)

\frac{6 ( u ^{4} + 1 )}{u ^{2}} \cdot 2 u^{2}

Multiply fractions:

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

= \frac{6 ( u ^{4} + 1 ) \cdot 2 u ^{2}}{u ^{2}}

Cancel the common factor:

u^{2}

= 6 (u^{4} + 1) \cdot 2

Multiply the numbers:

6 \cdot 2 = 12

= 12 (u^{4} + 1)

Simplify

\frac{7 ( u ^{2} - 1 )}{2 u} \cdot 2 u^{2} : 7 u (u^{2} - 1)

\frac{7 ( u ^{2} - 1 )}{2 u} \cdot 2 u^{2}

Multiply fractions:

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

= \frac{7 ( u ^{2} - 1 ) \cdot 2 u ^{2}}{2 u}

Cancel the common factor:

2

= \frac{7 ( u ^{2} - 1 ) u ^{2}}{u}

Cancel the common factor:

u

= 7 u (u^{2} - 1)

Simplify

- 24 \cdot 2 u^{2} : - 48 u^{2}

- 24 \cdot 2 u^{2}

Multiply the numbers:

24 \cdot 2 = 48

= - 48 u^{2}

Simplify

0 \cdot 2 u^{2} : 0

0 \cdot 2 u^{2}

Apply rule

0 \cdot a = 0

= 0

12 (u^{4} + 1) + 7 u (u^{2} - 1) - 48 u^{2} = 0

12 (u^{4} + 1) + 7 u (u^{2} - 1) - 48 u^{2} = 0

12 (u^{4} + 1) + 7 u (u^{2} - 1) - 48 u^{2} = 0

Solve

12 (u^{4} + 1) + 7 u (u^{2} - 1) - 48 u^{2} = 0 : u \approx 1.73025 \dots, u \approx 0.45623 \dots, u \approx - 0.57794 \dots, u \approx - 2.19187 \dots

12 (u^{4} + 1) + 7 u (u^{2} - 1) - 48 u^{2} = 0

Expand

12 (u^{4} + 1) + 7 u (u^{2} - 1) - 48 u^{2} : 12 u^{4} + 12 + 7 u^{3} - 7 u - 48 u^{2}

12 (u^{4} + 1) + 7 u (u^{2} - 1) - 48 u^{2}

Expand

12 (u^{4} + 1) : 12 u^{4} + 12

12 (u^{4} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 12, b = u^{4}, c = 1

= 12 u^{4} + 12 \cdot 1

Multiply the numbers:

12 \cdot 1 = 12

= 12 u^{4} + 12

= 12 u^{4} + 12 + 7 u (u^{2} - 1) - 48 u^{2}

Expand

7 u (u^{2} - 1) : 7 u^{3} - 7 u

7 u (u^{2} - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 7 u, b = u^{2}, c = 1

= 7 u u^{2} - 7 u \cdot 1

= 7 u^{2} u - 7 \cdot 1 \cdot u

Simplify

7 u^{2} u - 7 \cdot 1 \cdot u : 7 u^{3} - 7 u

7 u^{2} u - 7 \cdot 1 \cdot u

7 u^{2} u = 7 u^{3}

7 u^{2} u

Apply exponent rule:

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

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

= 7 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 7 u^{3}

7 \cdot 1 \cdot u = 7 u

7 \cdot 1 \cdot u

Multiply the numbers:

7 \cdot 1 = 7

= 7 u

= 7 u^{3} - 7 u

= 7 u^{3} - 7 u

= 12 u^{4} + 12 + 7 u^{3} - 7 u - 48 u^{2}

12 u^{4} + 12 + 7 u^{3} - 7 u - 48 u^{2} = 0

Write in the standard form

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

12 u^{4} + 7 u^{3} - 48 u^{2} - 7 u + 12 = 0

Find one solution for

12 u^{4} + 7 u^{3} - 48 u^{2} - 7 u + 12 = 0

using Newton-Raphson:

u \approx 1.73025 \dots

12 u^{4} + 7 u^{3} - 48 u^{2} - 7 u + 12 = 0

Newton-Raphson Approximation Definition

f (u) = 12 u^{4} + 7 u^{3} - 48 u^{2} - 7 u + 12

Find

f^{^{'}} (u) : 48 u^{3} + 21 u^{2} - 96 u - 7

\frac{d}{d u} (12 u^{4} + 7 u^{3} - 48 u^{2} - 7 u + 12)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (12 u^{4}) + \frac{d}{d u} (7 u^{3}) - \frac{d}{d u} (48 u^{2}) - \frac{d}{d u} (7 u) + \frac{d}{d u} (12)

\frac{d}{d u} (12 u^{4}) = 48 u^{3}

\frac{d}{d u} (12 u^{4})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 12 \frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 12 \cdot 4 u^{4 - 1}

Simplify

= 48 u^{3}

\frac{d}{d u} (7 u^{3}) = 21 u^{2}

\frac{d}{d u} (7 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 7 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 7 \cdot 3 u^{3 - 1}

Simplify

= 21 u^{2}

\frac{d}{d u} (48 u^{2}) = 96 u

\frac{d}{d u} (48 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 48 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 48 \cdot 2 u^{2 - 1}

Simplify

= 96 u

\frac{d}{d u} (7 u) = 7

\frac{d}{d u} (7 u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 7 \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 7 \cdot 1

Simplify

= 7

\frac{d}{d u} (12) = 0

\frac{d}{d u} (12)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 48 u^{3} + 21 u^{2} - 96 u - 7 + 0

Simplify

= 48 u^{3} + 21 u^{2} - 96 u - 7

Let

u_{0} = 2

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.79925 \dots : Δ u_{1} = 0.20074 \dots

f (u_{0}) = 12 \cdot 2^{4} + 7 \cdot 2^{3} - 48 \cdot 2^{2} - 7 \cdot 2 + 12 = 54

f^{^{'}} (u_{0}) = 48 \cdot 2^{3} + 21 \cdot 2^{2} - 96 \cdot 2 - 7 = 269

u_{1} = 1.79925 \dots

Δ u_{1} = ∣ 1.79925 \dots - 2 ∣ = 0.20074 \dots

Δ u_{1} = 0.20074 \dots

u_{2} = 1.73639 \dots : Δ u_{2} = 0.06285 \dots

f (u_{1}) = 12 \cdot 1.79925 \dots^{4} + 7 \cdot 1.79925 \dots^{3} - 48 \cdot 1.79925 \dots^{2} - 7 \cdot 1.79925 \dots + 12 = 10.55028 \dots

f^{^{'}} (u_{1}) = 48 \cdot 1.79925 \dots^{3} + 21 \cdot 1.79925 \dots^{2} - 96 \cdot 1.79925 \dots - 7 = 167.84443 \dots

u_{2} = 1.73639 \dots

Δ u_{2} = ∣ 1.73639 \dots - 1.79925 \dots ∣ = 0.06285 \dots

Δ u_{2} = 0.06285 \dots

u_{3} = 1.73031 \dots : Δ u_{3} = 0.00608 \dots

f (u_{2}) = 12 \cdot 1.73639 \dots^{4} + 7 \cdot 1.73639 \dots^{3} - 48 \cdot 1.73639 \dots^{2} - 7 \cdot 1.73639 \dots + 12 = 0.85758 \dots

f^{^{'}} (u_{2}) = 48 \cdot 1.73639 \dots^{3} + 21 \cdot 1.73639 \dots^{2} - 96 \cdot 1.73639 \dots - 7 = 140.92085 \dots

u_{3} = 1.73031 \dots

Δ u_{3} = ∣ 1.73031 \dots - 1.73639 \dots ∣ = 0.00608 \dots

Δ u_{3} = 0.00608 \dots

u_{4} = 1.73025 \dots : Δ u_{4} = 0.00005 \dots

f (u_{3}) = 12 \cdot 1.73031 \dots^{4} + 7 \cdot 1.73031 \dots^{3} - 48 \cdot 1.73031 \dots^{2} - 7 \cdot 1.73031 \dots + 12 = 0.00759 \dots

f^{^{'}} (u_{3}) = 48 \cdot 1.73031 \dots^{3} + 21 \cdot 1.73031 \dots^{2} - 96 \cdot 1.73031 \dots - 7 = 138.42910 \dots

u_{4} = 1.73025 \dots

Δ u_{4} = ∣ 1.73025 \dots - 1.73031 \dots ∣ = 0.00005 \dots

Δ u_{4} = 0.00005 \dots

u_{5} = 1.73025 \dots : Δ u_{5} = 4.43112 E - 9

f (u_{4}) = 12 \cdot 1.73025 \dots^{4} + 7 \cdot 1.73025 \dots^{3} - 48 \cdot 1.73025 \dots^{2} - 7 \cdot 1.73025 \dots + 12 = 6.13297 E - 7

f^{^{'}} (u_{4}) = 48 \cdot 1.73025 \dots^{3} + 21 \cdot 1.73025 \dots^{2} - 96 \cdot 1.73025 \dots - 7 = 138.40674 \dots

u_{5} = 1.73025 \dots

Δ u_{5} = ∣ 1.73025 \dots - 1.73025 \dots ∣ = 4.43112 E - 9

Δ u_{5} = 4.43112 E - 9

u \approx 1.73025 \dots

Apply long division:

\frac{12 u ^{4} + 7 u ^{3} - 48 u ^{2} - 7 u + 12}{u - 1.73025 \dots} = 12 u^{3} + 27.76310 \dots u^{2} + 0.03734 \dots u - 6.93537 \dots

12 u^{3} + 27.76310 \dots u^{2} + 0.03734 \dots u - 6.93537 \dots \approx 0

Find one solution for

12 u^{3} + 27.76310 \dots u^{2} + 0.03734 \dots u - 6.93537 \dots = 0

using Newton-Raphson:

u \approx 0.45623 \dots

12 u^{3} + 27.76310 \dots u^{2} + 0.03734 \dots u - 6.93537 \dots = 0

Newton-Raphson Approximation Definition

f (u) = 12 u^{3} + 27.76310 \dots u^{2} + 0.03734 \dots u - 6.93537 \dots

Find

f^{^{'}} (u) : 36 u^{2} + 55.52620 \dots u + 0.03734 \dots

\frac{d}{d u} (12 u^{3} + 27.76310 \dots u^{2} + 0.03734 \dots u - 6.93537 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (12 u^{3}) + \frac{d}{d u} (27.76310 \dots u^{2}) + \frac{d}{d u} (0.03734 \dots u) - \frac{d}{d u} (6.93537 \dots)

\frac{d}{d u} (12 u^{3}) = 36 u^{2}

\frac{d}{d u} (12 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 12 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 12 \cdot 3 u^{3 - 1}

Simplify

= 36 u^{2}

\frac{d}{d u} (27.76310 \dots u^{2}) = 55.52620 \dots u

\frac{d}{d u} (27.76310 \dots u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 27.76310 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 27.76310 \dots \cdot 2 u^{2 - 1}

Simplify

= 55.52620 \dots u

\frac{d}{d u} (0.03734 \dots u) = 0.03734 \dots

\frac{d}{d u} (0.03734 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 0.03734 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 0.03734 \dots \cdot 1

Simplify

= 0.03734 \dots

\frac{d}{d u} (6.93537 \dots) = 0

\frac{d}{d u} (6.93537 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 36 u^{2} + 55.52620 \dots u + 0.03734 \dots - 0

Simplify

= 36 u^{2} + 55.52620 \dots u + 0.03734 \dots

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.6410682029 : Δ u_{1} = 0.3589317971

f (u_{0}) = 12 \cdot 1^{3} + 27.76310 \dots \cdot 1^{2} + 0.03734 \dots \cdot 1 - 6.93537 \dots = 32.86507 \dots

f^{^{'}} (u_{0}) = 36 \cdot 1^{2} + 55.52620 \dots \cdot 1 + 0.03734 \dots = 91.56355 \dots

u_{1} = 0.6410682029

Δ u_{1} = ∣ 0.6410682029 - 1 ∣ = 0.3589317971

Δ u_{1} = 0.3589317971

u_{2} = 0.48917 \dots : Δ u_{2} = 0.15189 \dots

f (u_{1}) = 12 \cdot 0.641068202 9^{3} + 27.76310 \dots \cdot 0.641068202 9^{2} + 0.03734 \dots \cdot 0.6410682029 - 6.93537 \dots = 7.65982 \dots

f^{^{'}} (u_{1}) = 36 \cdot 0.641068202 9^{2} + 55.52620 \dots \cdot 0.6410682029 + 0.03734 \dots = 50.42829 \dots

u_{2} = 0.48917 \dots

Δ u_{2} = ∣ 0.48917 \dots - 0.6410682029 ∣ = 0.15189 \dots

Δ u_{2} = 0.15189 \dots

u_{3} = 0.45759 \dots : Δ u_{3} = 0.03157 \dots

f (u_{2}) = 12 \cdot 0.48917 \dots^{3} + 27.76310 \dots \cdot 0.48917 \dots^{2} + 0.03734 \dots \cdot 0.48917 \dots - 6.93537 \dots = 1.13097 \dots

f^{^{'}} (u_{2}) = 36 \cdot 0.48917 \dots^{2} + 55.52620 \dots \cdot 0.48917 \dots + 0.03734 \dots = 35.81369 \dots

u_{3} = 0.45759 \dots

Δ u_{3} = ∣ 0.45759 \dots - 0.48917 \dots ∣ = 0.03157 \dots

Δ u_{3} = 0.03157 \dots

u_{4} = 0.45623 \dots : Δ u_{4} = 0.00136 \dots

f (u_{3}) = 12 \cdot 0.45759 \dots^{3} + 27.76310 \dots \cdot 0.45759 \dots^{2} + 0.03734 \dots \cdot 0.45759 \dots - 6.93537 \dots = 0.04487 \dots

f^{^{'}} (u_{3}) = 36 \cdot 0.45759 \dots^{2} + 55.52620 \dots \cdot 0.45759 \dots + 0.03734 \dots = 32.98387 \dots

u_{4} = 0.45623 \dots

Δ u_{4} = ∣ 0.45623 \dots - 0.45759 \dots ∣ = 0.00136 \dots

Δ u_{4} = 0.00136 \dots

u_{5} = 0.45623 \dots : Δ u_{5} = 2.49018 E - 6

f (u_{4}) = 12 \cdot 0.45623 \dots^{3} + 27.76310 \dots \cdot 0.45623 \dots^{2} + 0.03734 \dots \cdot 0.45623 \dots - 6.93537 \dots = 0.00008 \dots

f^{^{'}} (u_{4}) = 36 \cdot 0.45623 \dots^{2} + 55.52620 \dots \cdot 0.45623 \dots + 0.03734 \dots = 32.86358 \dots

u_{5} = 0.45623 \dots

Δ u_{5} = ∣ 0.45623 \dots - 0.45623 \dots ∣ = 2.49018 E - 6

Δ u_{5} = 2.49018 E - 6

u_{6} = 0.45623 \dots : Δ u_{6} = 8.33775 E - 12

f (u_{5}) = 12 \cdot 0.45623 \dots^{3} + 27.76310 \dots \cdot 0.45623 \dots^{2} + 0.03734 \dots \cdot 0.45623 \dots - 6.93537 \dots = 2.74007 E - 10

f^{^{'}} (u_{5}) = 36 \cdot 0.45623 \dots^{2} + 55.52620 \dots \cdot 0.45623 \dots + 0.03734 \dots = 32.86336 \dots

u_{6} = 0.45623 \dots

Δ u_{6} = ∣ 0.45623 \dots - 0.45623 \dots ∣ = 8.33775 E - 12

Δ u_{6} = 8.33775 E - 12

u \approx 0.45623 \dots

Apply long division:

\frac{12 u ^{3} + 27.76310 \dots u ^{2} + 0.03734 \dots u - 6.93537 \dots}{u - 0.45623 \dots} = 12 u^{2} + 33.23786 \dots u + 15.20147 \dots

12 u^{2} + 33.23786 \dots u + 15.20147 \dots \approx 0

Find one solution for

12 u^{2} + 33.23786 \dots u + 15.20147 \dots = 0

using Newton-Raphson:

u \approx - 0.57794 \dots

12 u^{2} + 33.23786 \dots u + 15.20147 \dots = 0

Newton-Raphson Approximation Definition

f (u) = 12 u^{2} + 33.23786 \dots u + 15.20147 \dots

Find

f^{^{'}} (u) : 24 u + 33.23786 \dots

\frac{d}{d u} (12 u^{2} + 33.23786 \dots u + 15.20147 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (12 u^{2}) + \frac{d}{d u} (33.23786 \dots u) + \frac{d}{d u} (15.20147 \dots)

\frac{d}{d u} (12 u^{2}) = 24 u

\frac{d}{d u} (12 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 12 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 12 \cdot 2 u^{2 - 1}

Simplify

= 24 u

\frac{d}{d u} (33.23786 \dots u) = 33.23786 \dots

\frac{d}{d u} (33.23786 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 33.23786 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 33.23786 \dots \cdot 1

Simplify

= 33.23786 \dots

\frac{d}{d u} (15.20147 \dots) = 0

\frac{d}{d u} (15.20147 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 24 u + 33.23786 \dots + 0

Simplify

= 24 u + 33.23786 \dots

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.45735 \dots : Δ u_{1} = 0.45735 \dots

f (u_{0}) = 12 \cdot 0^{2} + 33.23786 \dots \cdot 0 + 15.20147 \dots = 15.20147 \dots

f^{^{'}} (u_{0}) = 24 \cdot 0 + 33.23786 \dots = 33.23786 \dots

u_{1} = - 0.45735 \dots

Δ u_{1} = ∣ - 0.45735 \dots - 0 ∣ = 0.45735 \dots

Δ u_{1} = 0.45735 \dots

u_{2} = - 0.57010 \dots : Δ u_{2} = 0.11275 \dots

f (u_{1}) = 12 (- 0.45735 \dots)^{2} + 33.23786 \dots (- 0.45735 \dots) + 15.20147 \dots = 2.51007 \dots

f^{^{'}} (u_{1}) = 24 (- 0.45735 \dots) + 33.23786 \dots = 22.26136 \dots

u_{2} = - 0.57010 \dots

Δ u_{2} = ∣ - 0.57010 \dots - (- 0.45735 \dots) ∣ = 0.11275 \dots

Δ u_{2} = 0.11275 \dots

u_{3} = - 0.57791 \dots : Δ u_{3} = 0.00780 \dots

f (u_{2}) = 12 (- 0.57010 \dots)^{2} + 33.23786 \dots (- 0.57010 \dots) + 15.20147 \dots = 0.15256 \dots

f^{^{'}} (u_{2}) = 24 (- 0.57010 \dots) + 33.23786 \dots = 19.55525 \dots

u_{3} = - 0.57791 \dots

Δ u_{3} = ∣ - 0.57791 \dots - (- 0.57010 \dots) ∣ = 0.00780 \dots

Δ u_{3} = 0.00780 \dots

u_{4} = - 0.57794 \dots : Δ u_{4} = 0.00003 \dots

f (u_{3}) = 12 (- 0.57791 \dots)^{2} + 33.23786 \dots (- 0.57791 \dots) + 15.20147 \dots = 0.00073 \dots

f^{^{'}} (u_{3}) = 24 (- 0.57791 \dots) + 33.23786 \dots = 19.36801 \dots

u_{4} = - 0.57794 \dots

Δ u_{4} = ∣ - 0.57794 \dots - (- 0.57791 \dots) ∣ = 0.00003 \dots

Δ u_{4} = 0.00003 \dots

u_{5} = - 0.57794 \dots : Δ u_{5} = 8.81165 E - 10

f (u_{4}) = 12 (- 0.57794 \dots)^{2} + 33.23786 \dots (- 0.57794 \dots) + 15.20147 \dots = 1.70656 E - 8

f^{^{'}} (u_{4}) = 24 (- 0.57794 \dots) + 33.23786 \dots = 19.36711 \dots

u_{5} = - 0.57794 \dots

Δ u_{5} = ∣ - 0.57794 \dots - (- 0.57794 \dots) ∣ = 8.81165 E - 10

Δ u_{5} = 8.81165 E - 10

u \approx - 0.57794 \dots

Apply long division:

\frac{12 u ^{2} + 33.23786 \dots u + 15.20147 \dots}{u + 0.57794 \dots} = 12 u + 26.30249 \dots

12 u + 26.30249 \dots \approx 0

u \approx - 2.19187 \dots

The solutions are

u \approx 1.73025 \dots, u \approx 0.45623 \dots, u \approx - 0.57794 \dots, u \approx - 2.19187 \dots

u \approx 1.73025 \dots, u \approx 0.45623 \dots, u \approx - 0.57794 \dots, u \approx - 2.19187 \dots

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

12 \frac{u ^{2} + u ^{- 2}}{2} + 7 \frac{u - u ^{- 1}}{2} - 24

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

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

u = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u \approx 1.73025 \dots, u \approx 0.45623 \dots, u \approx - 0.57794 \dots, u \approx - 2.19187 \dots

u \approx 1.73025 \dots, u \approx 0.45623 \dots, u \approx - 0.57794 \dots, u \approx - 2.19187 \dots

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 1.73025 \dots : x = ln (1.73025 \dots)

e^{x} = 1.73025 \dots

Apply exponent rules

e^{x} = 1.73025 \dots

f (x) = g (x)

, then

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

ln (e^{x}) = ln (1.73025 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (1.73025 \dots)

x = ln (1.73025 \dots)

Solve

e^{x} = 0.45623 \dots : x = ln (0.45623 \dots)

e^{x} = 0.45623 \dots

Apply exponent rules

e^{x} = 0.45623 \dots

f (x) = g (x)

, then

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

ln (e^{x}) = ln (0.45623 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (0.45623 \dots)

x = ln (0.45623 \dots)

Solve

e^{x} = - 0.57794 \dots :

No Solution for

x \in R

e^{x} = - 0.57794 \dots

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

Solve

e^{x} = - 2.19187 \dots :

No Solution for

x \in R

e^{x} = - 2.19187 \dots

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (1.73025 \dots), x = ln (0.45623 \dots)

x = ln (1.73025 \dots), x = ln (0.45623 \dots)

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 12cosh(2x)+7sinh(x)-24=0 ?
The general solution for 12cosh(2x)+7sinh(x)-24=0 is x=ln(1.73025…),x=ln(0.45623…)