Solve x^2 + y^2 - 4 x = 0

Identify the conic

Find the standard equation of the circle

Find the radius of the circle

Find the center of the circle

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

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Solve the equation

Solve for x

Solve for y

x = 2 + 4 - y^{2} x = 2 - 4 - 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{- x + 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{- y ^{2} - x ^{2} + 4 x - 4}{y ^{3}}

Calculate

x^{2} + y^{2} - 4 x = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} + y^{2} - 4 x) = \frac{d}{d x} (0)

Calculate the derivative

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2 x + 2 y \frac{d y}{d x} - 4 = \frac{d}{d x} (0)

Calculate the derivative

2 x + 2 y \frac{d y}{d x} - 4 = 0

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

2 y \frac{d y}{d x} = 0 - (2 x - 4)

Subtract the terms

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Calculate the derivative

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

Calculate

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

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

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

Solution

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

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Rewrite the equation

r = 0 r = 4 cos (θ)

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Identify the conic

Solve the equation

Testing for symmetry

Find the first derivative

Find the second derivative

Rewrite the equation

Graph