Solve (x^4)2 (y-4)2=25

Question

Solve the equation

Solve for x

Solve for y

x = - \frac{4 100 ( y - 4 ) ^{3}}{2 ∣ y - 4 ∣}, y \neq = 4 x = \frac{4 100 ( y - 4 ) ^{3}}{2 ∣ y - 4 ∣}, y \neq = 4

Evaluate

x^{4} \times 2 (y - 4) \times 2 = 25

Multiply

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4 x^{4} (y - 4) = 25

Rewrite the expression

(4 y - 16) x^{4} = 25

Divide both sides

\frac{( 4 y - 16 ) x ^{4}}{4 y - 16} = \frac{25}{4 y - 16}

Divide the numbers

x^{4} = \frac{25}{4 y - 16}

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm 4 \frac{25}{4 y - 16}

Simplify the expression

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x = \pm \frac{4 1600 ( y - 4 ) ^{3}}{4 ∣ y - 4 ∣}

Separate the equation into 2 possible cases

x = \frac{4 1600 ( y - 4 ) ^{3}}{4 ∣ y - 4 ∣} x = - \frac{4 1600 ( y - 4 ) ^{3}}{4 ∣ y - 4 ∣}

Calculate

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x = \frac{4 100 y ^{3} - 1200 y ^{2} + 4800 y - 6400}{2 ∣ y - 4 ∣} x = - \frac{4 1600 ( y - 4 ) ^{3}}{4 ∣ y - 4 ∣}

Calculate

More Steps

x = \frac{4 100 y ^{3} - 1200 y ^{2} + 4800 y - 6400}{2 ∣ y - 4 ∣} x = - \frac{4 100 y ^{3} - 1200 y ^{2} + 4800 y - 6400}{2 ∣ y - 4 ∣}

Calculate

{x = - \frac{4 100 y ^{3} - 1200 y ^{2} + 4800 y - 6400}{2 ∣ y - 4 ∣} y \neq = 4 {x = \frac{4 100 y ^{3} - 1200 y ^{2} + 4800 y - 6400}{2 ∣ y - 4 ∣} y \neq = 4

Simplify

x = - \frac{4 100 y ^{3} - 1200 y ^{2} + 4800 y - 6400}{2 ∣ y - 4 ∣}, y \neq = 4 x = \frac{4 100 ( y - 4 ) ^{3}}{2 ∣ y - 4 ∣}, y \neq = 4

Solution

x = - \frac{4 100 ( y - 4 ) ^{3}}{2 ∣ y - 4 ∣}, y \neq = 4 x = \frac{4 100 ( y - 4 ) ^{3}}{2 ∣ y - 4 ∣}, y \neq = 4

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Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{- 4 y + 16}{x}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{20 y - 80}{x ^{2}}

Calculate

(x^{4}) 2 (y - 4) 2 = 25

Simplify the expression

4 x^{4} (y - 4) = 25

Take the derivative of both sides

\frac{d}{d x} (4 x^{4} (y - 4)) = \frac{d}{d x} (25)

Calculate the derivative

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16 x^{3} y - 64 x^{3} + 4 x^{4} \frac{d y}{d x} = \frac{d}{d x} (25)

Calculate the derivative

16 x^{3} y - 64 x^{3} + 4 x^{4} \frac{d y}{d x} = 0

Move the expression to the right-hand side and change its sign

4 x^{4} \frac{d y}{d x} = 0 - (16 x^{3} y - 64 x^{3})

Subtract the terms

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4 x^{4} \frac{d y}{d x} = - 16 x^{3} y + 64 x^{3}

Divide both sides

\frac{4 x ^{4} \frac{d y}{d x}}{4 x ^{4}} = \frac{- 16 x ^{3} y + 64 x ^{3}}{4 x ^{4}}

Divide the numbers

\frac{d y}{d x} = \frac{- 16 x ^{3} y + 64 x ^{3}}{4 x ^{4}}

Divide the numbers

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\frac{d y}{d x} = \frac{- 4 y + 16}{x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{- 4 y + 16}{x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{- 4 y + 16}{x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( - 4 y + 16 ) \times x - ( - 4 y + 16 ) \times \frac{d}{d x} ( x )}{x ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 \frac{d y}{d x} \times x - ( - 4 y + 16 ) \times \frac{d}{d x} ( x )}{x ^{2}}

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 \frac{d y}{d x} \times x - ( - 4 y + 16 ) \times 1}{x ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 x \frac{d y}{d x} - ( - 4 y + 16 ) \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 x \frac{d y}{d x} - ( - 4 y + 16 )}{x ^{2}}

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 x \frac{d y}{d x} + 4 y - 16}{x ^{2}}

Use equation \frac{d y}{d x} = \frac{- 4 y + 16}{x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 x \times \frac{- 4 y + 16}{x} + 4 y - 16}{x ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{20 y - 80}{x ^{2}}

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