Solve x^2+y^2+x=0

Question

Identify the conic

Find the standard equation of the circle

Find the radius of the circle

Find the center of the circle

(x + \frac{1}{2})^{2} + y^{2} = \frac{1}{4}

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Solve the equation

Solve for x

Solve for y

x = \frac{- 1 + 1 - 4 y ^{2}}{2} x = - \frac{1 + 1 - 4 y ^{2}}{2}

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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{2 x + 1}{2 y}

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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{4 y ^{2} + 4 x ^{2} + 4 x + 1}{4 y ^{3}}

Calculate

x^{2} + y^{2} + x = 0

Take the derivative of both sides

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

Calculate the derivative

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2 x + 2 y \frac{d y}{d x} + 1 = \frac{d}{d x} (0)

Calculate the derivative

2 x + 2 y \frac{d y}{d x} + 1 = 0

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

2 y \frac{d y}{d x} = 0 - (2 x + 1)

Subtract the terms

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

Divide both sides

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

Divide the numbers

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

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

\frac{d y}{d x} = - \frac{2 x + 1}{2 y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{2 x + 1}{2 y})

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Calculate the derivative

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

Calculate

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

Calculate

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

Calculate

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

Calculate

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

Use equation \frac{d y}{d x} = - \frac{2 x + 1}{2 y} to substitute

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

Solution

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\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y ^{2} + 4 x ^{2} + 4 x + 1}{4 y ^{3}}

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Rewrite the equation

r = 0 r = - cos (θ)

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Graph