Solve -9x^2-36x-2y=18

Function

Find the x -intercept/zero

Find the y-intercept

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

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Solve the equation

Solve for x

Solve for y

x = \frac{- 6 + 18 - 2 y}{3} x = - \frac{6 + 18 - 2 y}{3}

Evaluate

- 9 x^{2} - 36 x - 2 y = 18

Move the expression to the left side

- 9 x^{2} - 36 x - 2 y - 18 = 0

Multiply both sides

9 x^{2} + 36 x + 2 y + 18 = 0

Substitute a= 9,b= 36 and c= 2 y + 18 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{- 36 \pm 3 6 ^{2} - 4 \times 9 ( 2 y + 18 )}{2 \times 9}

Simplify the expression

x = \frac{- 36 \pm 3 6 ^{2} - 4 \times 9 ( 2 y + 18 )}{18}

Simplify the expression

More Steps

x = \frac{- 36 \pm 648 - 72 y}{18}

Simplify the radical expression

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x = \frac{- 36 \pm 6 18 - 2 y}{18}

Separate the equation into 2 possible cases

x = \frac{- 36 + 6 18 - 2 y}{18} x = \frac{- 36 - 6 18 - 2 y}{18}

Simplify the expression

More Steps

x = \frac{- 6 + 18 - 2 y}{3} x = \frac{- 36 - 6 18 - 2 y}{18}

Solution

More Steps

x = \frac{- 6 + 18 - 2 y}{3} x = - \frac{6 + 18 - 2 y}{3}

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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 + 2)^{2} = - \frac{2}{9} (y - 9)

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Rewrite the equation

r = - \frac{18 cos ( θ ) + sin ( θ ) + 161 cos ^{2} ( θ ) + 18 sin ( 2 θ ) + 1}{9 cos ^{2} ( θ )} r = \frac{- 18 cos ( θ ) - sin ( θ ) + 161 cos ^{2} ( θ ) + 18 sin ( 2 θ ) + 1}{9 cos ^{2} ( θ )}

Evaluate

- 9 x^{2} - 36 x - 2 y = 18

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

- 9 (cos (θ) \times r)^{2} - 36 cos (θ) \times r - 2 sin (θ) \times r = 18

Factor the expression

- 9 cos^{2} (θ) \times r^{2} + (- 36 cos (θ) - 2 sin (θ)) r = 18

Subtract the terms

- 9 cos^{2} (θ) \times r^{2} + (- 36 cos (θ) - 2 sin (θ)) r - 18 = 18 - 18

Evaluate

- 9 cos^{2} (θ) \times r^{2} + (- 36 cos (θ) - 2 sin (θ)) r - 18 = 0

Solve using the quadratic formula

r = \frac{36 cos ( θ ) + 2 sin ( θ ) \pm ( - 36 cos ( θ ) - 2 sin ( θ ) ) ^{2} - 4 ( - 9 cos ^{2} ( θ ) ) ( - 18 )}{- 18 cos ^{2} ( θ )}

Simplify

r = \frac{36 cos ( θ ) + 2 sin ( θ ) \pm 644 cos ^{2} ( θ ) + 72 sin ( 2 θ ) + 4}{- 18 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{36 cos ( θ ) + 2 sin ( θ ) + 644 cos ^{2} ( θ ) + 72 sin ( 2 θ ) + 4}{- 18 cos ^{2} ( θ )} r = \frac{36 cos ( θ ) + 2 sin ( θ ) - 644 cos ^{2} ( θ ) + 72 sin ( 2 θ ) + 4}{- 18 cos ^{2} ( θ )}

Evaluate

More Steps

r = - \frac{18 cos ( θ ) + sin ( θ ) + 161 cos ^{2} ( θ ) + 18 sin ( 2 θ ) + 1}{9 cos ^{2} ( θ )} r = \frac{36 cos ( θ ) + 2 sin ( θ ) - 644 cos ^{2} ( θ ) + 72 sin ( 2 θ ) + 4}{- 18 cos ^{2} ( θ )}

Solution

More Steps

r = - \frac{18 cos ( θ ) + sin ( θ ) + 161 cos ^{2} ( θ ) + 18 sin ( 2 θ ) + 1}{9 cos ^{2} ( θ )} r = \frac{- 18 cos ( θ ) - sin ( θ ) + 161 cos ^{2} ( θ ) + 18 sin ( 2 θ ) + 1}{9 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} = - 9 x - 18

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Function

Solve the equation

Testing for symmetry

Identify the conic

Rewrite the equation

Find the first derivative

Graph