Solve 2(x^4)^(2)-6 = y

Question

Function

Find the x -intercept/zero

Find the y-intercept

x_{1} = - 83, x_{2} = 83

Evaluate

2 (x^{4})^{2} - 6 = y

To find the x -intercept,set y =0

2 (x^{4})^{2} - 6 = 0

Simplify

More Steps

Evaluate

2 (x^{4})^{2} - 6

Multiply the exponents

2 x^{4 \times 2} - 6

Multiply the numbers

2 x^{8} - 6

2 x^{8} - 6 = 0

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

2 x^{8} = 0 + 6

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

2 x^{8} = 6

Divide both sides

\frac{2 x ^{8}}{2} = \frac{6}{2}

Divide the numbers

x^{8} = \frac{6}{2}

Divide the numbers

More Steps

Evaluate

\frac{6}{2}

Reduce the numbers

\frac{3}{1}

Calculate

3

x^{8} = 3

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

x = \pm 83

Separate the equation into 2 possible cases

x = 83 x = - 83

Solution

x_{1} = - 83, x_{2} = 83

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{8 128 y + 768}{2} x = - \frac{8 128 y + 768}{2}

Evaluate

2 (x^{4})^{2} - 6 = y

Simplify

More Steps

Evaluate

2 (x^{4})^{2} - 6

Multiply the exponents

2 x^{4 \times 2} - 6

Multiply the numbers

2 x^{8} - 6

2 x^{8} - 6 = y

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

2 x^{8} = y + 6

Divide both sides

\frac{2 x ^{8}}{2} = \frac{y + 6}{2}

Divide the numbers

x^{8} = \frac{y + 6}{2}

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

x = \pm 8 \frac{y + 6}{2}

Simplify the expression

More Steps

Evaluate

8 \frac{y + 6}{2}

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

\frac{8 y + 6}{8 2}

Multiply by the Conjugate

\frac{8 y + 6 \times 8 2 ^{7}}{8 2 \times 8 2 ^{7}}

Calculate

\frac{8 y + 6 \times 8 2 ^{7}}{2}

Calculate

More Steps

Evaluate

8 y + 6 \times 8 2^{7}

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

8 (y + 6) \times 2^{7}

Calculate the product

8 128 y + 768

\frac{8 128 y + 768}{2}

x = \pm \frac{8 128 y + 768}{2}

Solution

x = \frac{8 128 y + 768}{2} x = - \frac{8 128 y + 768}{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

2 (x^{4})^{2} - 6 = y

Simplify the expression

2 x^{8} - 6 = y

To test if the graph of 2 x^{8} - 6 = y is symmetry with respect to the origin,substitute -x for x and -y for y

2 (- x)^{8} - 6 = - y

Evaluate

2 x^{8} - 6 = - y

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} = 16 x^{7}

Calculate

2 (x^{4})^{2} - 6 = y

Simplify the expression

2 x^{8} - 6 = y

Take the derivative of both sides

\frac{d}{d x} (2 x^{8} - 6) = \frac{d}{d x} (y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 x^{8} - 6)

Use differentiation rules

\frac{d}{d x} (2 x^{8}) + \frac{d}{d x} (- 6)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 x^{8})

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

2 \times \frac{d}{d x} (x^{8})

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

2 \times 8 x^{7}

Multiply the terms

16 x^{7}

16 x^{7} + \frac{d}{d x} (- 6)

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

16 x^{7} + 0

Evaluate

16 x^{7}

16 x^{7} = \frac{d}{d x} (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}

16 x^{7} = \frac{d y}{d x}

Solution

\frac{d y}{d x} = 16 x^{7}

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}} = 112 x^{6}

Calculate

2 (x^{4})^{2} - 6 = y

Simplify the expression

2 x^{8} - 6 = y

Take the derivative of both sides

\frac{d}{d x} (2 x^{8} - 6) = \frac{d}{d x} (y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 x^{8} - 6)

Use differentiation rules

\frac{d}{d x} (2 x^{8}) + \frac{d}{d x} (- 6)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 x^{8})

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

2 \times \frac{d}{d x} (x^{8})

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

2 \times 8 x^{7}

Multiply the terms

16 x^{7}

16 x^{7} + \frac{d}{d x} (- 6)

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

16 x^{7} + 0

Evaluate

16 x^{7}

16 x^{7} = \frac{d}{d x} (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}

16 x^{7} = \frac{d y}{d x}

Swap the sides of the equation

\frac{d y}{d x} = 16 x^{7}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (16 x^{7})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (16 x^{7})

Simplify

\frac{d ^{2} y}{d x ^{2}} = 16 \times \frac{d}{d x} (x^{7})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = 16 \times 7 x^{6}

Solution

\frac{d ^{2} y}{d x ^{2}} = 112 x^{6}

Show Solution

Function

Solve the equation

Testing for symmetry

Find the first derivative

Find the second derivative

Graph