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

Question

Function

Find the x -intercept/zero

Find the y-intercept

x_{1} = 0, x_{2} = 1

Evaluate

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

To find the x -intercept,set y =0

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

Any expression multiplied by 0 equals 0

x^{2} - 0 - x = 0

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

x^{2} - x = 0

Factor the expression

More Steps

Evaluate

x^{2} - x

Rewrite the expression

x \times x - x

Factor out x from the expression

x (x - 1)

x (x - 1) = 0

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

x = 0 x - 1 = 0

Solve the equation for x

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Evaluate

x - 1 = 0

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

x = 0 + 1

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

x = 1

x = 0 x = 1

Solution

x_{1} = 0, x_{2} = 1

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{1 + 1 + 16 y}{2} x = \frac{1 - 1 + 16 y}{2}

Evaluate

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

Rewrite in standard form

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

Substitute a= 1,b= - 1 and c= - 4 y into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{1 \pm ( - 1 ) ^{2} - 4 ( - 4 y )}{2}

Simplify the expression

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Evaluate

(- 1)^{2} - 4 (- 4 y)

Evaluate the power

1 - 4 (- 4 y)

Multiply the numbers

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Evaluate

4 (- 4 y)

Rewrite the expression

- 4 \times 4 y

Multiply the terms

- 16 y

1 - (- 16 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

1 + 16 y

x = \frac{1 \pm 1 + 16 y}{2}

Solution

x = \frac{1 + 1 + 16 y}{2} x = \frac{1 - 1 + 16 y}{2}

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

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

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

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

Evaluate

More Steps

Evaluate

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

Multiply the numbers

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

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

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

Rewrite the expression

x^{2} + 4 y + x

x^{2} + 4 y + x = 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 - \frac{1}{2})^{2} = 4 (y + \frac{1}{16})

Evaluate

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

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

x^{2} - x = 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

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

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

x^{2} - x = 4 y

To complete the square, the same value needs to be added to both sides

x^{2} - x + \frac{1}{4} = 4 y + \frac{1}{4}

Use a^{2} - 2 ab + b^{2} = (a - b)^{2} to factor the expression

(x - \frac{1}{2})^{2} = 4 y + \frac{1}{4}

Solution

(x - \frac{1}{2})^{2} = 4 (y + \frac{1}{16})

Show Solution

Rewrite the equation

r = 0 r = 4 sin (θ) sec^{2} (θ) + sec (θ)

Evaluate

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

Factor the expression

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

Factor the expression

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

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

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

Solution

More Steps

Factor the expression

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

Subtract the terms

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

Evaluate

cos^{2} (θ) \times r = 4 sin (θ) + cos (θ)

Divide the terms

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

Simplify the expression

r = 4 sin (θ) sec^{2} (θ) + sec (θ)

r = 0 r = 4 sin (θ) sec^{2} (θ) + sec (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 - 1}{4}

Calculate

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

Take the derivative of both sides

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

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

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}{d x} (- x)

Evaluate the derivative

More Steps

Evaluate

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

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

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

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

- 1

2 x - 4 \frac{d y}{d x} - 1

2 x - 4 \frac{d y}{d x} - 1 = \frac{d}{d x} (0)

Calculate the derivative

2 x - 4 \frac{d y}{d x} - 1 = 0

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

- 4 \frac{d y}{d x} = 0 - (2 x - 1)

Subtract the terms

More Steps

Evaluate

0 - (2 x - 1)

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

0 - 2 x + 1

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

- 2 x + 1

- 4 \frac{d y}{d x} = - 2 x + 1

Change the signs on both sides of the equation

4 \frac{d y}{d x} = 2 x - 1

Divide both sides

\frac{4 \frac{d y}{d x}}{4} = \frac{2 x - 1}{4}

Solution

\frac{d y}{d x} = \frac{2 x - 1}{4}

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{1}{2}

Calculate

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

Take the derivative of both sides

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

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

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}{d x} (- x)

Evaluate the derivative

More Steps

Evaluate

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

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

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

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

- 1

2 x - 4 \frac{d y}{d x} - 1

2 x - 4 \frac{d y}{d x} - 1 = \frac{d}{d x} (0)

Calculate the derivative

2 x - 4 \frac{d y}{d x} - 1 = 0

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

- 4 \frac{d y}{d x} = 0 - (2 x - 1)

Subtract the terms

More Steps

Evaluate

0 - (2 x - 1)

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

0 - 2 x + 1

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

- 2 x + 1

- 4 \frac{d y}{d x} = - 2 x + 1

Change the signs on both sides of the equation

4 \frac{d y}{d x} = 2 x - 1

Divide both sides

\frac{4 \frac{d y}{d x}}{4} = \frac{2 x - 1}{4}

Divide the numbers

\frac{d y}{d x} = \frac{2 x - 1}{4}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{2 x - 1}{4})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{2 x - 1}{4})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 2 x - 1 )}{4}

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 x - 1)

Use differentiation rules

\frac{d}{d x} (2 x) + \frac{d}{d x} (- 1)

Evaluate the derivative

2 + \frac{d}{d x} (- 1)

Use \frac{d}{d x} (c) = 0 to find derivative

2 + 0

Evaluate

2

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

Solution

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