Solve y=x^2-x 1 / x^2 x x - Socratic AI (ScanMath)

Question

Function

Evaluate the derivative

Find the domain

Find the x -intercept/zero

y^{'} = 2 x - 1

Evaluate

y = x^{2} - x \times \frac{1}{x ^{2}} \times x \times x

Simplify

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Evaluate

x^{2} - x \times \frac{1}{x ^{2}} \times x \times x

Multiply

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Evaluate

x \times \frac{1}{x ^{2}} \times x \times x

Multiply the terms with the same base by adding their exponents

x^{1 + 1 + 1} \times \frac{1}{x ^{2}}

Add the numbers

x^{3} \times \frac{1}{x ^{2}}

Cancel out the common factor x^{2}

x \times 1

Multiply the terms

x

x^{2} - x

y = x^{2} - x

Take the derivative of both sides

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

Use differentiation rule \frac{d}{d x} (f (x) \pm g (x)) = \frac{d}{d x} (f (x)) \pm \frac{d}{d x} (g (x))

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

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

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

Solution

y^{'} = 2 x - 1

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} - x \frac{1}{x ^{2}} xx

Simplify the expression

y = x^{2} - x

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

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

Simplify

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Evaluate

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

Rewrite the expression

(- x)^{2} + x

Rewrite the expression

x^{2} + x

- y = x^{2} + x

Change the signs both sides

y = - x^{2} - x

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} = y + \frac{1}{4}

Evaluate

y = x^{2} - x \times \frac{1}{x ^{2}} \times x \times x

Simplify

More Steps

Evaluate

x^{2} - x \times \frac{1}{x ^{2}} \times x \times x

Multiply

More Steps

Evaluate

x \times \frac{1}{x ^{2}} \times x \times x

Multiply the terms with the same base by adding their exponents

x^{1 + 1 + 1} \times \frac{1}{x ^{2}}

Add the numbers

x^{3} \times \frac{1}{x ^{2}}

Cancel out the common factor x^{2}

x \times 1

Multiply the terms

x

x^{2} - x

y = x^{2} - x

Swap the sides of the equation

x^{2} - x = y

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

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

Solution

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

Show Solution

Solve the equation

Solve for x

Solve for y

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

Evaluate

y = x^{2} - x \times \frac{1}{x ^{2}} \times x \times x

Simplify

More Steps

Evaluate

x^{2} - x \times \frac{1}{x ^{2}} \times x \times x

Multiply

More Steps

Evaluate

x \times \frac{1}{x ^{2}} \times x \times x

Multiply the terms with the same base by adding their exponents

x^{1 + 1 + 1} \times \frac{1}{x ^{2}}

Add the numbers

x^{3} \times \frac{1}{x ^{2}}

Cancel out the common factor x^{2}

x \times 1

Multiply the terms

x

x^{2} - x

y = x^{2} - x

Swap the sides of the equation

x^{2} - x = y

Add the same value to both sides

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

Evaluate

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

Evaluate

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

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

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

Simplify the expression

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Evaluate

\frac{1 + 4 y}{4}

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

\frac{1 + 4 y}{4}

Simplify the radical expression

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Evaluate

4

Write the number in exponential form with the base of 2

2^{2}

Reduce the index of the radical and exponent with 2

2

\frac{1 + 4 y}{2}

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

Separate the equation into 2 possible cases

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

Calculate

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Evaluate

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

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

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

Write all numerators above the common denominator

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

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

Solution

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Evaluate

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

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

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

Write all numerators above the common denominator

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

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

Show Solution

Rewrite the equation

r = 0 r = \frac{sin ( θ ) + cos ( θ )}{cos ^{2} ( θ )}

Evaluate

y = x^{2} - x \times \frac{1}{x ^{2}} \times x \times x

Simplify

More Steps

Evaluate

x^{2} - x \times \frac{1}{x ^{2}} \times x \times x

Multiply

More Steps

Evaluate

x \times \frac{1}{x ^{2}} \times x \times x

Multiply the terms with the same base by adding their exponents

x^{1 + 1 + 1} \times \frac{1}{x ^{2}}

Add the numbers

x^{3} \times \frac{1}{x ^{2}}

Cancel out the common factor x^{2}

x \times 1

Multiply the terms

x

x^{2} - x

y = x^{2} - x

Move the expression to the left side

y - x^{2} + x = 0

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

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

Factor the expression

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

Factor the expression

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

Solution

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Factor the expression

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

Subtract the terms

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

Evaluate

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

Divide the terms

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

r = 0 r = \frac{sin ( θ ) + cos ( θ )}{cos ^{2} ( θ )}

Show Solution

Y=x^2 X 1 / x^2 x x

Function

Testing for symmetry

Identify the conic

Solve the equation

Rewrite the equation

Graph