Solve y=x^2-4y - ScanMath

Question

y = x^{2} - 4 y

Function

Find the x -intercept/zero

Find the y-intercept

x = 0

Evaluate

y = x^{2} - 4 y

To find the x -intercept,set y =0

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

Any expression multiplied by 0 equals 0

0 = x^{2} - 0

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

0 = x^{2}

Swap the sides of the equation

x^{2} = 0

Solution

x = 0

Show Solution

Solve the equation

Solve for x

Solve for y

x = 5 y x = - 5 y

Evaluate

y = x^{2} - 4 y

Swap the sides of the equation

x^{2} - 4 y = y

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

x^{2} = y + 4 y

Add the terms

x^{2} = 5 y

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

x = \pm 5 y

Solution

x = 5 y x = - 5 y

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

y = x^{2} - 4 y

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

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

Evaluate

More Steps

Evaluate

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

Multiply the numbers

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

Rewrite the expression

(- x)^{2} + 4 y

Rewrite the expression

x^{2} + 4 y

- y = x^{2} + 4 y

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} = 5 y

Evaluate

y = x^{2} - 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

y - x^{2} = - 4 y

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

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

Add or subtract the terms

More Steps

Evaluate

- 4 y - y

Collect like terms by calculating the sum or difference of their coefficients

(- 4 - 1) y

Subtract the numbers

- 5 y

- x^{2} = - 5 y

Multiply both sides of the equation by - 1

- x^{2} (- 1) = - 5 y (- 1)

Multiplying or dividing an even number of negative terms equals a positive

x^{2} = - 5 y (- 1)

Solution

x^{2} = 5 y

Show Solution

Rewrite the equation

r = 0 r = 5 sin (θ) sec^{2} (θ)

Evaluate

y = x^{2} - 4 y

Move the expression to the left side

5 y - x^{2} = 0

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

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

Factor the expression

- cos^{2} (θ) \times r^{2} + 5 sin (θ) \times r = 0

Factor the expression

r (- cos^{2} (θ) \times r + 5 sin (θ)) = 0

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

r = 0 - cos^{2} (θ) \times r + 5 sin (θ) = 0

Solution

More Steps

Factor the expression

- cos^{2} (θ) \times r + 5 sin (θ) = 0

Subtract the terms

- cos^{2} (θ) \times r + 5 sin (θ) - 5 sin (θ) = 0 - 5 sin (θ)

Evaluate

- cos^{2} (θ) \times r = - 5 sin (θ)

Divide the terms

r = \frac{5 sin ( θ )}{cos ^{2} ( θ )}

Simplify the expression

r = 5 sin (θ) sec^{2} (θ)

r = 0 r = 5 sin (θ) sec^{2} (θ)

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{2 x}{5}

Calculate

y = x^{2} - 4 y

Take the derivative of both sides

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

\frac{d y}{d x}

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

2 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}

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

\frac{d y}{d x} = 2 x - 4 \frac{d y}{d x}

Move the variable to the left side

\frac{d y}{d x} + 4 \frac{d y}{d x} = 2 x

Add the terms

More Steps

Evaluate

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

Collect like terms by calculating the sum or difference of their coefficients

(1 + 4) \frac{d y}{d x}

Add the numbers

5 \frac{d y}{d x}

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

Divide both sides

\frac{5 \frac{d y}{d x}}{5} = \frac{2 x}{5}

Solution

\frac{d y}{d x} = \frac{2 x}{5}

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{2}{5}

Calculate

y = x^{2} - 4 y

Take the derivative of both sides

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

\frac{d y}{d x}

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

2 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}

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

\frac{d y}{d x} = 2 x - 4 \frac{d y}{d x}

Move the variable to the left side

\frac{d y}{d x} + 4 \frac{d y}{d x} = 2 x

Add the terms

More Steps

Evaluate

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

Collect like terms by calculating the sum or difference of their coefficients

(1 + 4) \frac{d y}{d x}

Add the numbers

5 \frac{d y}{d x}

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

Divide both sides

\frac{5 \frac{d y}{d x}}{5} = \frac{2 x}{5}

Divide the numbers

\frac{d y}{d x} = \frac{2 x}{5}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{2 x}{5})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{2 x}{5})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 2 x )}{5}

Solution

More Steps

Evaluate

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

Simplify

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

Rewrite the expression

2 \times 1

Any expression multiplied by 1 remains the same

2

\frac{d ^{2} y}{d x ^{2}} = \frac{2}{5}

Show Solution

Function

Solve the equation

Testing for symmetry

Identify the conic

Rewrite the equation

Find the first derivative

Find the second derivative

Graph