Solve x=3y^2+2 - ScanMath

Question

Function

Find the vertex

Find the axis of symmetry

Find the domain

(0, 2)

Evaluate

x = 3 y^{2} + 2

Find the y -coordinate of the vertex by substituting a= 3 and b= 0 into y = - \frac{b}{2 a}

y = - \frac{0}{2 \times 3}

Solve the equation for y

y = 0

Find the y-coordinate of the vertex by evaluating the function for y = 0

x = 3 \times 0^{2} + 2

Calculate

More Steps

Evaluate

3 \times 0^{2} + 2

Calculate

3 \times 0 + 2

Any expression multiplied by 0 equals 0

0 + 2

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

2

x = 2

Solution

(0, 2)

Show Solution

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

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

Evaluate

x = 3 y^{2} + 2

Swap the sides of the equation

3 y^{2} + 2 = x

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

3 y^{2} = x - 2

Multiply both sides of the equation by \frac{1}{3}

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

Multiply the terms

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

Multiply the terms

More Steps

Evaluate

(x - 2) \times \frac{1}{3}

Apply the distributive property

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

Use the commutative property to reorder the terms

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

Multiply the numbers

\frac{1}{3} x - \frac{2}{3}

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

Solution

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

Show Solution

Solve the equation

y = \frac{3 x - 6}{3} y = - \frac{3 x - 6}{3}

Evaluate

x = 3 y^{2} + 2

Swap the sides of the equation

3 y^{2} + 2 = x

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

3 y^{2} = x - 2

Divide both sides

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

Divide the numbers

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

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

y = \pm \frac{x - 2}{3}

Simplify the expression

More Steps

Evaluate

\frac{x - 2}{3}

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

\frac{x - 2}{3}

Multiply by the Conjugate

\frac{x - 2 \times 3}{3 \times 3}

Calculate

\frac{x - 2 \times 3}{3}

Calculate

More Steps

Evaluate

x - 2 \times 3

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

(x - 2) \times 3

Calculate the product

3 x - 6

\frac{3 x - 6}{3}

y = \pm \frac{3 x - 6}{3}

Solution

y = \frac{3 x - 6}{3} y = - \frac{3 x - 6}{3}

Show Solution

Rewrite the equation

r = \frac{cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )} r = \frac{cos ( θ ) + 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )}

Evaluate

x = 3 y^{2} + 2

Move the expression to the left side

x - 3 y^{2} = 2

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

cos (θ) \times r - 3 (sin (θ) \times r)^{2} = 2

Factor the expression

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

Subtract the terms

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

Evaluate

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

Solve using the quadratic formula

r = \frac{- cos ( θ ) \pm cos ^{2} ( θ ) - 4 ( - 3 sin ^{2} ( θ ) ) ( - 2 )}{- 6 sin ^{2} ( θ )}

Simplify

r = \frac{- cos ( θ ) \pm 25 cos ^{2} ( θ ) - 24}{- 6 sin ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- cos ( θ ) + 25 cos ^{2} ( θ ) - 24}{- 6 sin ^{2} ( θ )} r = \frac{- cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{- 6 sin ^{2} ( θ )}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

r = \frac{cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )} r = \frac{- cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{- 6 sin ^{2} ( θ )}

Solution

r = \frac{cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )} r = \frac{cos ( θ ) + 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )}

Show Solution

Function

Identify the conic

Solve the equation

Rewrite the equation

Graph