What are the solutions to the equation 0.4(ue^{-4})^2=0.2((ue^{-4}-u)/4) ?

The solutions to the equation 0.4(ue^{-4})^2=0.2((ue^{-4}-u)/4) are u=0,u=-(e^8-e^4)/8

Find the zeros of 0.4(ue^{-4})^2=0.2((ue^{-4}-u)/4)

The zeros of 0.4(ue^{-4})^2=0.2((ue^{-4}-u)/4) are u=0,u=-(e^8-e^4)/8

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0.4 (u e^{- 4})^{2} = 0.2 (\frac{u e ^{- 4} - u}{4})

u = 0, u = - \frac{e ^{8} - e ^{4}}{8}

Solution steps

0.4 (u e^{- 4})^{2} = 0.2 (\frac{u e ^{- 4} - u}{4})

Multiply both sides by

10

\frac{4}{e ^{8}} u^{2} = \frac{( - e ^{4} + 1 ) u}{2 e ^{4}}

Find Least Common Multiplier of

e^{8}, 2 e^{4} : 2 e^{8}

Multiply by LCM=

2 e^{8}

\frac{4}{e ^{8}} u^{2} \cdot 2 e^{8} = \frac{( - e ^{4} + 1 ) u}{2 e ^{4}} \cdot 2 e^{8}

Simplify

8 u^{2} = e^{4} (- e^{4} + 1) u

Expand

e^{4} (- e^{4} + 1) u : - e^{8} u + e^{4} u

8 u^{2} = - e^{8} u + e^{4} u

Subtract

- e^{8} u + e^{4} u

from both sides

8 u^{2} - (- e^{8} u + e^{4} u) = - e^{8} u + e^{4} u - (- e^{8} u + e^{4} u)

Simplify

8 u^{2} + (e^{8} - e^{4}) u = 0

Solve with the quadratic formula

u_{1, 2} = \frac{- ( e ^{8} - e ^{4} ) \pm ( e ^{8} - e ^{4} ) ^{2} - 4 \cdot 8 \cdot 0}{2 \cdot 8}

(e^{8} - e^{4})^{2} - 4 \cdot 8 \cdot 0 = e^{8} - e^{4}

u_{1, 2} = \frac{- ( e ^{8} - e ^{4} ) \pm ( e ^{8} - e ^{4} )}{2 \cdot 8}

Separate the solutions

u_{1} = \frac{- ( e ^{8} - e ^{4} ) + e ^{8} - e ^{4}}{2 \cdot 8}, u_{2} = \frac{- ( e ^{8} - e ^{4} ) - ( e ^{8} - e ^{4} )}{2 \cdot 8}

u = \frac{- ( e ^{8} - e ^{4} ) + e ^{8} - e ^{4}}{2 \cdot 8} : 0

u = \frac{- ( e ^{8} - e ^{4} ) - ( e ^{8} - e ^{4} )}{2 \cdot 8} : - \frac{e ^{8} - e ^{4}}{8}

The solutions to the quadratic equation are:

u = 0, u = - \frac{e ^{8} - e ^{4}}{8}

x (b - 4) + z (b - 4)

\frac{4}{3} \cdot π \cdot 3^{3}

\frac{3}{a} + \frac{7}{b}

7 x^{2} - 14 x - 13 = 8

\frac{x ^{2} + x}{x ^{2} - 1} > 0

What are the solutions to the equation 0.4(ue^{-4})^2=0.2((ue^{-4}-u)/4) ?
The solutions to the equation 0.4(ue^{-4})^2=0.2((ue^{-4}-u)/4) are u=0,u=-(e^8-e^4)/8
Find the zeros of 0.4(ue^{-4})^2=0.2((ue^{-4}-u)/4)
The zeros of 0.4(ue^{-4})^2=0.2((ue^{-4}-u)/4) are u=0,u=-(e^8-e^4)/8