Solve -25x^216y^2-200x 160y-400=0

Question

Solve the equation

Solve for x

Solve for y

x = - \frac{40 + 1599}{y} x = \frac{- 40 + 1599}{y}

Evaluate

- 25 x^{2} \times 16 y^{2} - 200 x \times 160 y - 400 = 0

Simplify

More Steps

- 400 x^{2} y^{2} - 32000 x y - 400 = 0

Rewrite the expression

- 400 y^{2} x^{2} - 32000 y x - 400 = 0

Substitute a= - 400 y^{2},b= - 32000 y and c= - 400 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{32000 y \pm ( - 32000 y ) ^{2} - 4 ( - 400 y ^{2} ) ( - 400 )}{2 ( - 400 y ^{2} )}

Simplify the expression

x = \frac{32000 y \pm ( - 32000 y ) ^{2} - 4 ( - 400 y ^{2} ) ( - 400 )}{- 800 y ^{2}}

Simplify the expression

More Steps

x = \frac{32000 y \pm 3200 0 ^{2} y ^{2} - 640000 y ^{2}}{- 800 y ^{2}}

Simplify the radical expression

More Steps

x = \frac{32000 y \pm 800 1599 \times y}{- 800 y ^{2}}

Separate the equation into 2 possible cases

x = \frac{32000 y + 800 1599 \times y}{- 800 y ^{2}} x = \frac{32000 y - 800 1599 \times y}{- 800 y ^{2}}

Simplify the expression

More Steps

x = - \frac{40 + 1599}{y} x = \frac{32000 y - 800 1599 \times y}{- 800 y ^{2}}

Solution

More Steps

x = - \frac{40 + 1599}{y} x = \frac{- 40 + 1599}{y}

Show Solution

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{y}{x}

Calculate

- 25 x^{2} 16 y^{2} - 200 x 160 y - 400 = 0

Simplify the expression

- 400 x^{2} y^{2} - 32000 x y - 400 = 0

Take the derivative of both sides

\frac{d}{d x} (- 400 x^{2} y^{2} - 32000 x y - 400) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

- 800 x y^{2} - 800 x^{2} y \frac{d y}{d x} - 32000 y - 32000 x \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

- 800 x y^{2} - 800 x^{2} y \frac{d y}{d x} - 32000 y - 32000 x \frac{d y}{d x} = 0

Collect like terms by calculating the sum or difference of their coefficients

- 800 x y^{2} - 32000 y + (- 800 x^{2} y - 32000 x) \frac{d y}{d x} = 0

Move the constant to the right side

(- 800 x^{2} y - 32000 x) \frac{d y}{d x} = 0 + 800 x y^{2} + 32000 y

Removing 0 doesn't change the value,so remove it from the expression

(- 800 x^{2} y - 32000 x) \frac{d y}{d x} = 800 x y^{2} + 32000 y

Divide both sides

\frac{( - 800 x ^{2} y - 32000 x ) \frac{d y}{d x}}{- 800 x ^{2} y - 32000 x} = \frac{800 x y ^{2} + 32000 y}{- 800 x ^{2} y - 32000 x}

Divide the numbers

\frac{d y}{d x} = \frac{800 x y ^{2} + 32000 y}{- 800 x ^{2} y - 32000 x}

Solution

More Steps

\frac{d y}{d x} = - \frac{y}{x}

Show Solution

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{2 y}{x ^{2}}

Calculate

- 25 x^{2} 16 y^{2} - 200 x 160 y - 400 = 0

Simplify the expression

- 400 x^{2} y^{2} - 32000 x y - 400 = 0

Take the derivative of both sides

\frac{d}{d x} (- 400 x^{2} y^{2} - 32000 x y - 400) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

- 800 x y^{2} - 800 x^{2} y \frac{d y}{d x} - 32000 y - 32000 x \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

- 800 x y^{2} - 800 x^{2} y \frac{d y}{d x} - 32000 y - 32000 x \frac{d y}{d x} = 0

Collect like terms by calculating the sum or difference of their coefficients

- 800 x y^{2} - 32000 y + (- 800 x^{2} y - 32000 x) \frac{d y}{d x} = 0

Move the constant to the right side

(- 800 x^{2} y - 32000 x) \frac{d y}{d x} = 0 + 800 x y^{2} + 32000 y

Removing 0 doesn't change the value,so remove it from the expression

(- 800 x^{2} y - 32000 x) \frac{d y}{d x} = 800 x y^{2} + 32000 y

Divide both sides

\frac{( - 800 x ^{2} y - 32000 x ) \frac{d y}{d x}}{- 800 x ^{2} y - 32000 x} = \frac{800 x y ^{2} + 32000 y}{- 800 x ^{2} y - 32000 x}

Divide the numbers

\frac{d y}{d x} = \frac{800 x y ^{2} + 32000 y}{- 800 x ^{2} y - 32000 x}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{y}{x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{y}{x})

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times x - y \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{\frac{d y}{d x} \times x - y \times 1}{x ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{x \frac{d y}{d x} - y \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

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

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

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{2 y}{x ^{2}}

Show Solution