Solve x^2+y^2-3x+2y=5

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{3}{2})^{2} + (y + 1)^{2} = \frac{33}{4}

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Solve the equation

Solve for x

Solve for y

x = \frac{3 + 29 - 4 y ^{2} - 8 y}{2} x = \frac{3 - 29 - 4 y ^{2} - 8 y}{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 + 3}{2 y + 2}

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Find the second derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{2} - 8 y - 13 - 4 x ^{2} + 12 x}{4 y ^{3} + 12 y ^{2} + 12 y + 4}

Calculate

x^{2} + y^{2} - 3 x + 2 y = 5

Take the derivative of both sides

\frac{d}{d x} (x^{2} + y^{2} - 3 x + 2 y) = \frac{d}{d x} (5)

Calculate the derivative

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2 x + 2 y \frac{d y}{d x} - 3 + 2 \frac{d y}{d x} = \frac{d}{d x} (5)

Calculate the derivative

2 x + 2 y \frac{d y}{d x} - 3 + 2 \frac{d y}{d x} = 0

Simplify

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2 x - 3 + (2 y + 2) \frac{d y}{d x} = 0

Move the constant to the right side

(2 y + 2) \frac{d y}{d x} = 0 - (2 x - 3)

Subtract the terms

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(2 y + 2) \frac{d y}{d x} = - 2 x + 3

Divide both sides

\frac{( 2 y + 2 ) \frac{d y}{d x}}{2 y + 2} = \frac{- 2 x + 3}{2 y + 2}

Divide the numbers

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

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{- 2 x + 3}{2 y + 2})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{- 2 x + 3}{2 y + 2})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( - 2 x + 3 ) \times ( 2 y + 2 ) - ( - 2 x + 3 ) \times \frac{d}{d x} ( 2 y + 2 )}{( 2 y + 2 ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 2 ( 2 y + 2 ) - ( - 2 x + 3 ) \times \frac{d}{d x} ( 2 y + 2 )}{( 2 y + 2 ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 2 ( 2 y + 2 ) - ( - 2 x + 3 ) \times 2 \frac{d y}{d x}}{( 2 y + 2 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y - 4 - ( - 2 x + 3 ) \times 2 \frac{d y}{d x}}{( 2 y + 2 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y - 4 - ( - 4 x \frac{d y}{d x} + 6 \frac{d y}{d x} )}{( 2 y + 2 ) ^{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 + 4 x \frac{d y}{d x} - 6 \frac{d y}{d x}}{( 2 y + 2 ) ^{2}}

Calculate

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

Use equation \frac{d y}{d x} = \frac{- 2 x + 3}{2 y + 2} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- 2 y - 2 + 2 x \times \frac{- 2 x + 3}{2 y + 2} - 3 \times \frac{- 2 x + 3}{2 y + 2}}{2 ( y + 1 ) ^{2}}

Solution

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Calculate

\frac{- 2 y - 2 + 2 x \times \frac{- 2 x + 3}{2 y + 2} - 3 \times \frac{- 2 x + 3}{2 y + 2}}{2 ( y + 1 ) ^{2}}

Multiply the terms

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\frac{- 2 y - 2 + \frac{x ( - 2 x + 3 )}{y + 1} - 3 \times \frac{- 2 x + 3}{2 y + 2}}{2 ( y + 1 ) ^{2}}

Multiply the terms

\frac{- 2 y - 2 + \frac{x ( - 2 x + 3 )}{y + 1} - \frac{3 ( - 2 x + 3 )}{2 y + 2}}{2 ( y + 1 ) ^{2}}

Calculate the sum or difference

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\frac{\frac{- 4 y ^{2} - 8 y - 13 - 4 x ^{2} + 12 x}{2 ( y + 1 )}}{2 ( y + 1 ) ^{2}}

Multiply by the reciprocal

\frac{- 4 y ^{2} - 8 y - 13 - 4 x ^{2} + 12 x}{2 ( y + 1 )} \times \frac{1}{2 ( y + 1 ) ^{2}}

Multiply the terms

\frac{- 4 y ^{2} - 8 y - 13 - 4 x ^{2} + 12 x}{2 ( y + 1 ) \times 2 ( y + 1 ) ^{2}}

Multiply the terms

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\frac{- 4 y ^{2} - 8 y - 13 - 4 x ^{2} + 12 x}{4 ( y + 1 ) ^{3}}

Expand the expression

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\frac{- 4 y ^{2} - 8 y - 13 - 4 x ^{2} + 12 x}{4 y ^{3} + 12 y ^{2} + 12 y + 4}

\frac{d ^{2} y}{d x ^{2}} = \frac{- 4 y ^{2} - 8 y - 13 - 4 x ^{2} + 12 x}{4 y ^{3} + 12 y ^{2} + 12 y + 4}

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