Solve x x-3/2/2=y - Socratic AI (ScanMath)

Function

Find the x -intercept/zero

Find the y-intercept

x_{1} = - \frac{3}{2}, x_{2} = \frac{3}{2}

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Solve for x

Solve for y

x = \frac{4 y + 3}{2} x = - \frac{4 y + 3}{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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Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

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

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

Evaluate

x \times x - \frac{3}{2} \div 2 = y

Evaluate

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

Multiply both sides of the equation by LCD

(x^{2} - \frac{3}{4}) \times 4 = y \times 4

Simplify the equation

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4 x^{2} - 3 = y \times 4

Use the commutative property to reorder the terms

4 x^{2} - 3 = 4 y

Move the expression to the left side

4 x^{2} - 3 - 4 y = 0

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

4 (cos (θ) \times r)^{2} - 3 - 4 sin (θ) \times r = 0

Factor the expression

4 cos^{2} (θ) \times r^{2} - 4 sin (θ) \times r - 3 = 0

Solve using the quadratic formula

r = \frac{4 sin ( θ ) \pm ( - 4 sin ( θ ) ) ^{2} - 4 \times 4 cos ^{2} ( θ ) ( - 3 )}{8 cos ^{2} ( θ )}

Simplify

r = \frac{4 sin ( θ ) \pm 16 + 32 cos ^{2} ( θ )}{8 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{4 sin ( θ ) + 16 + 32 cos ^{2} ( θ )}{8 cos ^{2} ( θ )} r = \frac{4 sin ( θ ) - 16 + 32 cos ^{2} ( θ )}{8 cos ^{2} ( θ )}

Evaluate

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r = \frac{sin ( θ ) + 1 + 2 cos ^{2} ( θ )}{2 cos ^{2} ( θ )} r = \frac{4 sin ( θ ) - 16 + 32 cos ^{2} ( θ )}{8 cos ^{2} ( θ )}

Solution

More Steps

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

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