Solve 2x^2 + 2y^2 - 2x + 6y + 5 = 0

Solve the equation

Solve for x

Solve for y

x = \frac{1 + - 9 - 12 y - 4 y ^{2}}{2} x = \frac{1 - - 9 - 12 y - 4 y ^{2}}{2}

Evaluate

2 x^{2} + 2 y^{2} - 2 x + 6 y + 5 = 0

Rewrite the expression

2 x^{2} + 2 y^{2} + 6 y + 5 - 2 x = 0

Rewrite in standard form

2 x^{2} - 2 x + 2 y^{2} + 6 y + 5 = 0

Substitute a= 2,b= - 2 and c= 2 y^{2} + 6 y + 5 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{2 \pm ( - 2 ) ^{2} - 4 \times 2 ( 2 y ^{2} + 6 y + 5 )}{2 \times 2}

Simplify the expression

x = \frac{2 \pm ( - 2 ) ^{2} - 4 \times 2 ( 2 y ^{2} + 6 y + 5 )}{4}

Simplify the expression

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x = \frac{2 \pm - 36 - 16 y ^{2} - 48 y}{4}

Simplify the radical expression

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x = \frac{2 \pm 2 - 9 - 12 y - 4 y ^{2}}{4}

Separate the equation into 2 possible cases

x = \frac{2 + 2 - 9 - 12 y - 4 y ^{2}}{4} x = \frac{2 - 2 - 9 - 12 y - 4 y ^{2}}{4}

Simplify the expression

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x = \frac{1 + - 9 - 12 y - 4 y ^{2}}{2} x = \frac{2 - 2 - 9 - 12 y - 4 y ^{2}}{4}

Solution

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x = \frac{1 + - 9 - 12 y - 4 y ^{2}}{2} x = \frac{1 - - 9 - 12 y - 4 y ^{2}}{2}

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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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r = \frac{cos ( θ ) - 3 sin ( θ ) + - 9 + 8 sin ^{2} ( θ ) - 3 sin ( 2 θ )}{2} r = \frac{cos ( θ ) - 3 sin ( θ ) - - 9 + 8 sin ^{2} ( θ ) - 3 sin ( 2 θ )}{2}

Evaluate

2 x^{2} + 2 y^{2} - 2 x + 6 y + 5 = 0

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

2 (cos (θ) \times r)^{2} + 2 (sin (θ) \times r)^{2} - 2 cos (θ) \times r + 6 sin (θ) \times r + 5 = 0

Factor the expression

(2 cos^{2} (θ) + 2 sin^{2} (θ)) r^{2} + (- 2 cos (θ) + 6 sin (θ)) r + 5 = 0

Simplify the expression

2 r^{2} + (- 2 cos (θ) + 6 sin (θ)) r + 5 = 0

Solve using the quadratic formula

r = \frac{2 cos ( θ ) - 6 sin ( θ ) \pm ( - 2 cos ( θ ) + 6 sin ( θ ) ) ^{2} - 4 \times 2 \times 5}{4}

Simplify

r = \frac{2 cos ( θ ) - 6 sin ( θ ) \pm - 36 + 32 sin ^{2} ( θ ) - 12 sin ( 2 θ )}{4}

Separate the equation into 2 possible cases

r = \frac{2 cos ( θ ) - 6 sin ( θ ) + - 36 + 32 sin ^{2} ( θ ) - 12 sin ( 2 θ )}{4} r = \frac{2 cos ( θ ) - 6 sin ( θ ) - - 36 + 32 sin ^{2} ( θ ) - 12 sin ( 2 θ )}{4}

Evaluate

More Steps

r = \frac{cos ( θ ) - 3 sin ( θ ) + - 9 + 8 sin ^{2} ( θ ) - 3 sin ( 2 θ )}{2} r = \frac{2 cos ( θ ) - 6 sin ( θ ) - - 36 + 32 sin ^{2} ( θ ) - 12 sin ( 2 θ )}{4}

Solution

More Steps

r = \frac{cos ( θ ) - 3 sin ( θ ) + - 9 + 8 sin ^{2} ( θ ) - 3 sin ( 2 θ )}{2} r = \frac{cos ( θ ) - 3 sin ( θ ) - - 9 + 8 sin ^{2} ( θ ) - 3 sin ( 2 θ )}{2}

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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 + 3}

Calculate

2 x^{2} + 2 y^{2} - 2 x + 6 y + 5 = 0

Take the derivative of both sides

\frac{d}{d x} (2 x^{2} + 2 y^{2} - 2 x + 6 y + 5) = \frac{d}{d x} (0)

Calculate the derivative

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4 x + 4 y \frac{d y}{d x} - 2 + 6 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

4 x + 4 y \frac{d y}{d x} - 2 + 6 \frac{d y}{d x} = 0

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

4 x - 2 + (4 y + 6) \frac{d y}{d x} = 0

Move the constant to the right side

(4 y + 6) \frac{d y}{d x} = 0 - (4 x - 2)

Subtract the terms

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

Divide both sides

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

Divide the numbers

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

Solution

More Steps

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

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