Solve (x^2)^2 = -4/3 (y - 1)

Question

Function

Find the x -intercept/zero

Find the y-intercept

x_{1} = - \frac{4 108}{3}, x_{2} = \frac{4 108}{3}

Evaluate

(x^{2})^{2} = - \frac{4}{3} (y - 1)

To find the x -intercept,set y =0

(x^{2})^{2} = - \frac{4}{3} (0 - 1)

Simplify

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Evaluate

(x^{2})^{2}

Multiply the exponents

x^{2 \times 2}

Multiply the numbers

x^{4}

x^{4} = - \frac{4}{3} (0 - 1)

Simplify

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Evaluate

- \frac{4}{3} (0 - 1)

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

- \frac{4}{3} (- 1)

Multiplying or dividing an odd number of negative terms equals a negative

- (- \frac{4}{3})

Calculate

\frac{4}{3}

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

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

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

Simplify the expression

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Evaluate

4 \frac{4}{3}

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

\frac{4 4}{4 3}

Simplify the radical expression

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Evaluate

44

Write the number in exponential form with the base of 2

4 2^{2}

Reduce the index of the radical and exponent with 2

2

\frac{2}{4 3}

Multiply by the Conjugate

\frac{2 \times 4 3 ^{3}}{4 3 \times 4 3 ^{3}}

Simplify

\frac{2 \times 4 27}{4 3 \times 4 3 ^{3}}

Multiply the numbers

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Evaluate

2 \times 427

Use n a = mn a^{m} to expand the expression

4 2^{2} \times 427

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

4 2^{2} \times 27

Calculate the product

4108

\frac{4 108}{4 3 \times 4 3 ^{3}}

Multiply the numbers

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Evaluate

43 \times 4 3^{3}

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

4 3 \times 3^{3}

Calculate the product

4 3^{4}

Reduce the index of the radical and exponent with 4

3

\frac{4 108}{3}

x = \pm \frac{4 108}{3}

Separate the equation into 2 possible cases

x = \frac{4 108}{3} x = - \frac{4 108}{3}

Solution

x_{1} = - \frac{4 108}{3}, x_{2} = \frac{4 108}{3}

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{4 - 108 y + 108}{3} x = - \frac{4 - 108 y + 108}{3}

Evaluate

(x^{2})^{2} = - \frac{4}{3} (y - 1)

Simplify

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Evaluate

(x^{2})^{2}

Multiply the exponents

x^{2 \times 2}

Multiply the numbers

x^{4}

x^{4} = - \frac{4}{3} (y - 1)

Simplify

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Evaluate

- \frac{4}{3} (y - 1)

Multiply the terms

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Evaluate

\frac{4}{3} (y - 1)

Apply the distributive property

\frac{4}{3} y - \frac{4}{3} \times 1

Any expression multiplied by 1 remains the same

\frac{4}{3} y - \frac{4}{3}

- (\frac{4}{3} y - \frac{4}{3})

Calculate

- \frac{4}{3} y + \frac{4}{3}

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

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

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

Simplify the expression

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Evaluate

4 - \frac{4}{3} y + \frac{4}{3}

Factor the expression

4 \frac{4}{3} (- y + 1)

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

4 \frac{4}{3} \times 4 - y + 1

Evaluate the root

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Evaluate

4 \frac{4}{3}

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

\frac{4 4}{4 3}

Simplify the radical expression

\frac{2}{4 3}

Multiply by the Conjugate

\frac{2 \times 4 3 ^{3}}{4 3 \times 4 3 ^{3}}

Simplify

\frac{2 \times 4 27}{4 3 \times 4 3 ^{3}}

Multiply the numbers

\frac{4 108}{4 3 \times 4 3 ^{3}}

Multiply the numbers

\frac{4 108}{3}

\frac{4 108}{3} 4 - y + 1

Calculate the product

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Evaluate

4108 \times 4 - y + 1

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

4 108 (- y + 1)

Calculate the product

4 - 108 y + 108

\frac{4 - 108 y + 108}{3}

x = \pm \frac{4 - 108 y + 108}{3}

Solution

x = \frac{4 - 108 y + 108}{3} x = - \frac{4 - 108 y + 108}{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

(x^{2})^{2} = - \frac{4}{3} (y - 1)

Simplify the expression

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

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

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

Evaluate

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

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

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

Solution

Not symmetry with respect to the origin

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} = - 3 x^{3}

Calculate

(x^{2})^{2} = - \frac{4}{3} (y - 1)

Simplify the expression

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

Take the derivative of both sides

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

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

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

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Evaluate

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

Use differentiation rules

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

Evaluate the derivative

- \frac{4}{3} \frac{d y}{d x}

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

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

- \frac{4}{3} \frac{d y}{d x} + 0

Evaluate

- \frac{4}{3} \frac{d y}{d x}

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

Swap the sides of the equation

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

Change the signs on both sides of the equation

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

Multiply by the reciprocal

\frac{4}{3} \frac{d y}{d x} \times \frac{3}{4} = - 4 x^{3} \times \frac{3}{4}

Multiply

\frac{d y}{d x} = - 4 x^{3} \times \frac{3}{4}

Solution

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Evaluate

- 4 x^{3} \times \frac{3}{4}

Reduce the numbers

- x^{3} \times 3

Use the commutative property to reorder the terms

- 3 x^{3}

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

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}} = - 9 x^{2}

Calculate

(x^{2})^{2} = - \frac{4}{3} (y - 1)

Simplify the expression

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

Take the derivative of both sides

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

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

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

- \frac{4}{3} \frac{d y}{d x}

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

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

- \frac{4}{3} \frac{d y}{d x} + 0

Evaluate

- \frac{4}{3} \frac{d y}{d x}

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

Swap the sides of the equation

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

Change the signs on both sides of the equation

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

Multiply by the reciprocal

\frac{4}{3} \frac{d y}{d x} \times \frac{3}{4} = - 4 x^{3} \times \frac{3}{4}

Multiply

\frac{d y}{d x} = - 4 x^{3} \times \frac{3}{4}

Multiply

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Evaluate

- 4 x^{3} \times \frac{3}{4}

Reduce the numbers

- x^{3} \times 3

Use the commutative property to reorder the terms

- 3 x^{3}

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

Take the derivative of both sides

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

Calculate the derivative

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

Simplify

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

Rewrite the expression

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

Solution

\frac{d ^{2} y}{d x ^{2}} = - 9 x^{2}

Show Solution

Function

Solve the equation

Testing for symmetry

Find the first derivative

Find the second derivative

Graph