Solve x^2+(y-3\sqrt x^2)2=1

Function

Find the x -intercept/zero

Find the y-intercept

x_{1} = 3 - 10, x_{2} = 3 + 10

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Solve the equation

Solve for x

Solve for y

x = 3 + 10 - 2 y x = 3 - 10 - 2 y

Evaluate

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

Simplify

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

Move the expression to the left side

x^{2} + 2 (y - 3 x) - 1 = 0

Calculate

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x^{2} + 2 y - 6 x - 1 = 0

Simplify

x^{2} + 2 y - 1 - 6 x = 0

Rewrite in standard form

x^{2} - 6 x + 2 y - 1 = 0

Substitute a= 1,b= - 6 and c= 2 y - 1 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{6 \pm ( - 6 ) ^{2} - 4 ( 2 y - 1 )}{2}

Simplify the expression

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x = \frac{6 \pm 40 - 8 y}{2}

Simplify the radical expression

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x = \frac{6 \pm 2 10 - 2 y}{2}

Separate the equation into 2 possible cases

x = \frac{6 + 2 10 - 2 y}{2} x = \frac{6 - 2 10 - 2 y}{2}

Simplify the expression

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x = 3 + 10 - 2 y x = \frac{6 - 2 10 - 2 y}{2}

Solution

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x = 3 + 10 - 2 y x = 3 - 10 - 2 y

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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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Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

(x - 3)^{2} = - 2 (y - 5)

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Rewrite the equation

r = \frac{- sin ( θ ) + 3 cos ( θ ) + 1 + 9 cos ^{2} ( θ ) - 3 sin ( 2 θ )}{cos ^{2} ( θ )} r = \frac{- sin ( θ ) + 3 cos ( θ ) - 1 + 9 cos ^{2} ( θ ) - 3 sin ( 2 θ )}{cos ^{2} ( θ )}

Evaluate

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

Evaluate

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x^{2} + 2 y - 6 x = 1

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

(cos (θ) \times r)^{2} + 2 sin (θ) \times r - 6 cos (θ) \times r = 1

Factor the expression

cos^{2} (θ) \times r^{2} + (2 sin (θ) - 6 cos (θ)) r = 1

Subtract the terms

cos^{2} (θ) \times r^{2} + (2 sin (θ) - 6 cos (θ)) r - 1 = 1 - 1

Evaluate

cos^{2} (θ) \times r^{2} + (2 sin (θ) - 6 cos (θ)) r - 1 = 0

Solve using the quadratic formula

r = \frac{- 2 sin ( θ ) + 6 cos ( θ ) \pm ( 2 sin ( θ ) - 6 cos ( θ ) ) ^{2} - 4 cos ^{2} ( θ ) ( - 1 )}{2 cos ^{2} ( θ )}

Simplify

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

Separate the equation into 2 possible cases

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

Evaluate

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r = \frac{- sin ( θ ) + 3 cos ( θ ) + 1 + 9 cos ^{2} ( θ ) - 3 sin ( 2 θ )}{cos ^{2} ( θ )} r = \frac{- 2 sin ( θ ) + 6 cos ( θ ) - 4 + 36 cos ^{2} ( θ ) - 12 sin ( 2 θ )}{2 cos ^{2} ( θ )}

Solution

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r = \frac{- sin ( θ ) + 3 cos ( θ ) + 1 + 9 cos ^{2} ( θ ) - 3 sin ( 2 θ )}{cos ^{2} ( θ )} r = \frac{- sin ( θ ) + 3 cos ( θ ) - 1 + 9 cos ^{2} ( θ ) - 3 sin ( 2 θ )}{cos ^{2} ( θ )}

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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} = - x + 3

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Function

Solve the equation

Testing for symmetry

Identify the conic

Rewrite the equation

Find the first derivative

Graph