Solve x^2-4x 6=y - ScanMath

Question

Function

Find the x -intercept/zero

Find the y-intercept

x_{1} = 0, x_{2} = 24

Evaluate

x^{2} - 4 x \times 6 = y

To find the x -intercept,set y =0

x^{2} - 4 x \times 6 = 0

Multiply the terms

x^{2} - 24 x = 0

Factor the expression

More Steps

Evaluate

x^{2} - 24 x

Rewrite the expression

x \times x - x \times 24

Factor out x from the expression

x (x - 24)

x (x - 24) = 0

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

x = 0 x - 24 = 0

Solve the equation for x

More Steps

Evaluate

x - 24 = 0

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

x = 0 + 24

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

x = 24

x = 0 x = 24

Solution

x_{1} = 0, x_{2} = 24

Show Solution

Solve the equation

Solve for x

Solve for y

x = 12 + 144 + y x = 12 - 144 + y

Evaluate

x^{2} - 4 x \times 6 = y

Multiply the terms

x^{2} - 24 x = y

Move the expression to the left side

x^{2} - 24 x - y = 0

Substitute a= 1,b= - 24 and c= - y into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{24 \pm ( - 24 ) ^{2} - 4 ( - y )}{2}

Simplify the expression

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Evaluate

(- 24)^{2} - 4 (- y)

Use the commutative property to reorder the terms

(- 24)^{2} - (- 4 y)

Rewrite the expression

2 4^{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

2 4^{2} + 4 y

Evaluate the power

576 + 4 y

x = \frac{24 \pm 576 + 4 y}{2}

Simplify the radical expression

More Steps

Evaluate

576 + 4 y

Factor the expression

4 (144 + y)

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

4 \times 144 + y

Evaluate the root

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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

2 144 + y

x = \frac{24 \pm 2 144 + y}{2}

Separate the equation into 2 possible cases

x = \frac{24 + 2 144 + y}{2} x = \frac{24 - 2 144 + y}{2}

Simplify the expression

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Evaluate

x = \frac{24 + 2 144 + y}{2}

Divide the terms

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Evaluate

\frac{24 + 2 144 + y}{2}

Rewrite the expression

\frac{2 ( 12 + 144 + y )}{2}

Reduce the fraction

12 + 144 + y

x = 12 + 144 + y

x = 12 + 144 + y x = \frac{24 - 2 144 + y}{2}

Solution

More Steps

Evaluate

x = \frac{24 - 2 144 + y}{2}

Divide the terms

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Evaluate

\frac{24 - 2 144 + y}{2}

Rewrite the expression

\frac{2 ( 12 - 144 + y )}{2}

Reduce the fraction

12 - 144 + y

x = 12 - 144 + y

x = 12 + 144 + y x = 12 - 144 + 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

x^{2} - 4 x 6 = y

Simplify the expression

x^{2} - 24 x = y

To test if the graph of x^{2} - 24 x = y is symmetry with respect to the origin,substitute -x for x and -y for y

(- x)^{2} - 24 (- x) = - y

Evaluate

More Steps

Evaluate

(- x)^{2} - 24 (- x)

Multiply the numbers

(- x)^{2} - (- 24 x)

Rewrite the expression

(- x)^{2} + 24 x

Rewrite the expression

x^{2} + 24 x

x^{2} + 24 x = - 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 - 12)^{2} = y + 144

Evaluate

x^{2} - 4 x \times 6 = y

Calculate

x^{2} - 24 x = y

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

x^{2} - 24 x + 144 = y + 144

Solution

(x - 12)^{2} = y + 144

Show Solution

Rewrite the equation

r = 0 r = \frac{24 cos ( θ ) + sin ( θ )}{cos ^{2} ( θ )}

Evaluate

x^{2} - 4 x \times 6 = y

Evaluate

x^{2} - 24 x = y

Move the expression to the left side

x^{2} - 24 x - y = 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} - 24 cos (θ) \times r - sin (θ) \times r = 0

Factor the expression

cos^{2} (θ) \times r^{2} + (- 24 cos (θ) - sin (θ)) r = 0

Factor the expression

r (cos^{2} (θ) \times r - 24 cos (θ) - sin (θ)) = 0

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

r = 0 cos^{2} (θ) \times r - 24 cos (θ) - sin (θ) = 0

Solution

More Steps

Factor the expression

cos^{2} (θ) \times r - 24 cos (θ) - sin (θ) = 0

Subtract the terms

cos^{2} (θ) \times r - 24 cos (θ) - sin (θ) - (- 24 cos (θ) - sin (θ)) = 0 - (- 24 cos (θ) - sin (θ))

Evaluate

cos^{2} (θ) \times r = 24 cos (θ) + sin (θ)

Divide the terms

r = \frac{24 cos ( θ ) + sin ( θ )}{cos ^{2} ( θ )}

r = 0 r = \frac{24 cos ( θ ) + sin ( θ )}{cos ^{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} = 2 x - 24

Calculate

x^{2} - 4 x 6 = y

Simplify the expression

x^{2} - 24 x = y

Take the derivative of both sides

\frac{d}{d x} (x^{2} - 24 x) = \frac{d}{d x} (y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (x^{2} - 24 x)

Use differentiation rules

\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 24 x)

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

2 x + \frac{d}{d x} (- 24 x)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 24 x)

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

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

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

- 24 \times 1

Any expression multiplied by 1 remains the same

- 24

2 x - 24

2 x - 24 = \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}

2 x - 24 = \frac{d y}{d x}

Solution

\frac{d y}{d x} = 2 x - 24

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}} = 2

Calculate

x^{2} - 4 x 6 = y

Simplify the expression

x^{2} - 24 x = y

Take the derivative of both sides

\frac{d}{d x} (x^{2} - 24 x) = \frac{d}{d x} (y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (x^{2} - 24 x)

Use differentiation rules

\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 24 x)

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

2 x + \frac{d}{d x} (- 24 x)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 24 x)

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

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

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

- 24 \times 1

Any expression multiplied by 1 remains the same

- 24

2 x - 24

2 x - 24 = \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}

2 x - 24 = \frac{d y}{d x}

Swap the sides of the equation

\frac{d y}{d x} = 2 x - 24

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (2 x - 24)

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (2 x - 24)

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (2 x) + \frac{d}{d x} (- 24)

Evaluate the derivative

More Steps

Evaluate

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

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)

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

2 \times 1

Any expression multiplied by 1 remains the same

2

\frac{d ^{2} y}{d x ^{2}} = 2 + \frac{d}{d x} (- 24)

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

\frac{d ^{2} y}{d x ^{2}} = 2 + 0

Solution

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