Solve x x/2-6=y - ScanMath

Function

Find the x -intercept/zero

Find the y-intercept

x_{1} = - 23, x_{2} = 23

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Solve for x

Solve for y

x = 2 y + 12 x = - 2 y + 12

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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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Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

x^{2} = 2 (y + 6)

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r = \frac{sin ( θ ) + 1 + 11 cos ^{2} ( θ )}{cos ^{2} ( θ )} r = \frac{sin ( θ ) - 1 + 11 cos ^{2} ( θ )}{cos ^{2} ( θ )}

Evaluate

x \times \frac{x}{2} - 6 = y

Evaluate

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\frac{x ^{2}}{2} - 6 = y

Multiply both sides of the equation by LCD

(\frac{x ^{2}}{2} - 6) \times 2 = y \times 2

Simplify the equation

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x^{2} - 12 = y \times 2

Use the commutative property to reorder the terms

x^{2} - 12 = 2 y

Move the expression to the left side

x^{2} - 12 - 2 y = 0

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

(cos (θ) \times r)^{2} - 12 - 2 sin (θ) \times r = 0

Factor the expression

cos^{2} (θ) \times r^{2} - 2 sin (θ) \times r - 12 = 0

Solve using the quadratic formula

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

Simplify

r = \frac{2 sin ( θ ) \pm 4 + 44 cos ^{2} ( θ )}{2 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{2 sin ( θ ) + 4 + 44 cos ^{2} ( θ )}{2 cos ^{2} ( θ )} r = \frac{2 sin ( θ ) - 4 + 44 cos ^{2} ( θ )}{2 cos ^{2} ( θ )}

Evaluate

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r = \frac{sin ( θ ) + 1 + 11 cos ^{2} ( θ )}{cos ^{2} ( θ )} r = \frac{2 sin ( θ ) - 4 + 44 cos ^{2} ( θ )}{2 cos ^{2} ( θ )}

Solution

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r = \frac{sin ( θ ) + 1 + 11 cos ^{2} ( θ )}{cos ^{2} ( θ )} r = \frac{sin ( θ ) - 1 + 11 cos ^{2} ( θ )}{cos ^{2} ( θ )}

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = x

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Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = 1

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