Solve 3x^2-4y=0 - ScanMath

Question

Function

Find the x -intercept/zero

Find the y-intercept

x = 0

Evaluate

3 x^{2} - 4 y = 0

To find the x -intercept,set y =0

3 x^{2} - 4 \times 0 = 0

Any expression multiplied by 0 equals 0

3 x^{2} - 0 = 0

Removing 0 doesn't change the value,so remove it from the expression

3 x^{2} = 0

Rewrite the expression

x^{2} = 0

Solution

x = 0

Show Solution

Solve the equation

Solve for x

Solve for y

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

Evaluate

3 x^{2} - 4 y = 0

Move the expression to the right-hand side and change its sign

3 x^{2} = 0 + 4 y

Add the terms

3 x^{2} = 4 y

Divide both sides

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

Divide the numbers

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

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm \frac{4 y}{3}

Simplify the expression

More Steps

Evaluate

\frac{4 y}{3}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{4 y}{3}

Multiply by the Conjugate

\frac{4 y \times 3}{3 \times 3}

Calculate

\frac{4 y \times 3}{3}

Calculate

More Steps

Evaluate

4 y \times 3

The product of roots with the same index is equal to the root of the product

4 y \times 3

Calculate the product

12 y

Write the expression as a product where the root of one of the factors can be evaluated

4 \times 3 y

Write the number in exponential form with the base of 2

2^{2} \times 3 y

The root of a product is equal to the product of the roots of each factor

2^{2} \times 3 y

Reduce the index of the radical and exponent with 2

2 3 y

\frac{2 3 y}{3}

x = \pm \frac{2 3 y}{3}

Solution

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

Show Solution

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

Evaluate

3 x^{2} - 4 y = 0

To test if the graph of 3 x^{2} - 4 y = 0 is symmetry with respect to the origin,substitute -x for x and -y for y

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

Evaluate

More Steps

Evaluate

3 (- x)^{2} - 4 (- y)

Multiply the terms

3 x^{2} - 4 (- y)

Multiply the numbers

3 x^{2} - (- 4 y)

Rewrite the expression

3 x^{2} + 4 y

3 x^{2} + 4 y = 0

Solution

Not symmetry with respect to the origin

Show Solution

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

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

Evaluate

3 x^{2} - 4 y = 0

Move the expression to the right-hand side and change its sign

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

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

3 x^{2} = 0 + 4 y

Removing 0 doesn't change the value,so remove it from the expression

3 x^{2} = 4 y

Multiply both sides of the equation by \frac{1}{3}

3 x^{2} \times \frac{1}{3} = 4 y \times \frac{1}{3}

Multiply the terms

x^{2} = 4 y \times \frac{1}{3}

Solution

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

Show Solution

Rewrite the equation

r = 0 r = \frac{4 sin ( θ ) sec ^{2} ( θ )}{3}

Evaluate

3 x^{2} - 4 y = 0

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

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

Factor the expression

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

Factor the expression

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

When the product of factors equals 0,at least one factor is 0

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

Solution

More Steps

Factor the expression

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

Subtract the terms

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

Evaluate

3 cos^{2} (θ) \times r = 4 sin (θ)

Divide the terms

r = \frac{4 sin ( θ )}{3 cos ^{2} ( θ )}

Simplify the expression

r = \frac{4 sin ( θ ) sec ^{2} ( θ )}{3}

r = 0 r = \frac{4 sin ( θ ) sec ^{2} ( θ )}{3}

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{3 x}{2}

Calculate

3 x^{2} - 4 y = 0

Take the derivative of both sides

\frac{d}{d x} (3 x^{2} - 4 y) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

\frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 4 y)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 x^{2})

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

3 \times \frac{d}{d x} (x^{2})

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

3 \times 2 x

Multiply the terms

6 x

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

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 4 y)

Use differentiation rules

\frac{d}{d y} (- 4 y) \times \frac{d y}{d x}

Evaluate the derivative

- 4 \frac{d y}{d x}

6 x - 4 \frac{d y}{d x}

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

Calculate the derivative

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

Move the expression to the right-hand side and change its sign

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

Removing 0 doesn't change the value,so remove it from the expression

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

Change the signs on both sides of the equation

4 \frac{d y}{d x} = 6 x

Divide both sides

\frac{4 \frac{d y}{d x}}{4} = \frac{6 x}{4}

Divide the numbers

\frac{d y}{d x} = \frac{6 x}{4}

Solution

\frac{d y}{d x} = \frac{3 x}{2}

Show Solution

Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{3}{2}

Calculate

3 x^{2} - 4 y = 0

Take the derivative of both sides

\frac{d}{d x} (3 x^{2} - 4 y) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

\frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 4 y)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 x^{2})

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

3 \times \frac{d}{d x} (x^{2})

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

3 \times 2 x

Multiply the terms

6 x

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

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 4 y)

Use differentiation rules

\frac{d}{d y} (- 4 y) \times \frac{d y}{d x}

Evaluate the derivative

- 4 \frac{d y}{d x}

6 x - 4 \frac{d y}{d x}

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

Calculate the derivative

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

Move the expression to the right-hand side and change its sign

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

Removing 0 doesn't change the value,so remove it from the expression

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

Change the signs on both sides of the equation

4 \frac{d y}{d x} = 6 x

Divide both sides

\frac{4 \frac{d y}{d x}}{4} = \frac{6 x}{4}

Divide the numbers

\frac{d y}{d x} = \frac{6 x}{4}

Cancel out the common factor 2

\frac{d y}{d x} = \frac{3 x}{2}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{3 x}{2})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{3 x}{2})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 3 x )}{2}

Solution

More Steps

Evaluate

\frac{d}{d x} (3 x)

Simplify

3 \times \frac{d}{d x} (x)

Rewrite the expression

3 \times 1

Any expression multiplied by 1 remains the same

3

\frac{d ^{2} y}{d x ^{2}} = \frac{3}{2}

Show Solution

Function

Solve the equation

Testing for symmetry

Identify the conic

Rewrite the equation

Find the first derivative

Find the second derivative

Graph