Algebra Examples

$3 y - y^{2} = x^{2}$

Step 1

Since $x = r \cos (θ)$ , replace $x$ with $r \cos (θ)$ .

$3 y - y^{2} = {(r \cos (θ))}^{2}$

Step 2

Since $y = r \sin (θ)$ , replace $y$ with $r \sin (θ)$ .

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

Step 3

Solve for $r$ .

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Step 3.1

Subtract ${(r \cos (θ))}^{2}$ from both sides of the equation.

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

Step 3.2

Simplify the left side of the equation.

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Step 3.2.1

Simplify terms.

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Apply the product rule to $r \sin (θ)$ .

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

Step 3.2.1.1.2

Apply the product rule to $r \cos (θ)$ .

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

Step 3.2.1.2

Simplify with factoring out.

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Step 3.2.1.2.1

Factor $- r^{2}$ out of $- r^{2} \sin^{2} (θ)$ .

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

Step 3.2.1.2.2

Factor $- r^{2}$ out of $- r^{2} \cos^{2} (θ)$ .

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

Step 3.2.1.2.3

Factor $- r^{2}$ out of $- r^{2} (\sin^{2} (θ)) - r^{2} (\cos^{2} (θ))$ .

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

Step 3.2.2

Apply pythagorean identity.

$3 r \sin (θ) - r^{2} \cdot 1 = 0$

Step 3.2.3

Multiply $- 1$ by $1$ .

$3 r \sin (θ) - r^{2} = 0$

Step 3.3

Factor $r$ out of $3 r \sin (θ) - r^{2}$ .

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Step 3.3.1

Factor $r$ out of $3 r \sin (θ)$ .

$r (3 \sin (θ)) - r^{2} = 0$

Step 3.3.2

Factor $r$ out of $- r^{2}$ .

$r (3 \sin (θ)) + r (- r) = 0$

Step 3.3.3

Factor $r$ out of $r (3 \sin (θ)) + r (- r)$ .

$r (3 \sin (θ) - r) = 0$

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$r = 0$

$3 \sin (θ) - r = 0$

Step 3.5

Set $r$ equal to $0$ .

$r = 0$

Step 3.6

Set $3 \sin (θ) - r$ equal to $0$ and solve for $r$ .

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Step 3.6.1

Set $3 \sin (θ) - r$ equal to $0$ .

$3 \sin (θ) - r = 0$

Step 3.6.2

Solve $3 \sin (θ) - r = 0$ for $r$ .

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Step 3.6.2.1

Subtract $3 \sin (θ)$ from both sides of the equation.

$- r = - 3 \sin (θ)$

Step 3.6.2.2

Divide each term in $- r = - 3 \sin (θ)$ by $- 1$ and simplify.

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Step 3.6.2.2.1

Divide each term in $- r = - 3 \sin (θ)$ by $- 1$ .

$\frac{- r}{- 1} = \frac{- 3 \sin (θ)}{- 1}$

Step 3.6.2.2.2

Simplify the left side.

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Step 3.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{r}{1} = \frac{- 3 \sin (θ)}{- 1}$

Step 3.6.2.2.2.2

Divide $r$ by $1$ .

$r = \frac{- 3 \sin (θ)}{- 1}$

Step 3.6.2.2.3

Simplify the right side.

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Step 3.6.2.2.3.1

Move the negative one from the denominator of $\frac{- 3 \sin (θ)}{- 1}$ .

$r = - 1 \cdot (- 3 \sin (θ))$

Step 3.6.2.2.3.2

Rewrite $- 1 \cdot (- 3 \sin (θ))$ as $- (- 3 \sin (θ))$ .

$r = - (- 3 \sin (θ))$

Step 3.6.2.2.3.3

Multiply $- 3$ by $- 1$ .

$r = 3 \sin (θ)$

Step 3.7

The final solution is all the values that make $r (3 \sin (θ) - r) = 0$ true.

$r = 0$

$r = 3 \sin (θ)$

$r = 0$

$r = 3 \sin (θ)$