Algebra Examples

$r = 1 + 2 \sin (θ)$

Step 1

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

$r = 1 + 2 \frac{y}{r}$

Step 2

Multiply both sides by $r$ .

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Step 2.1

Multiply both sides by $r$ .

$r \cdot r = r (1 + 2 \frac{y}{r})$

Step 2.2

Simplify the left side.

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Step 2.2.1

Multiply $r$ by $r$ .

$r^{2} = r (1 + 2 \frac{y}{r})$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify $r (1 + 2 \frac{y}{r})$ .

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Step 2.3.1.1

Combine $2$ and $\frac{y}{r}$ .

$r^{2} = r (1 + \frac{2 y}{r})$

Step 2.3.1.2

Apply the distributive property.

$r^{2} = r \cdot 1 + r \frac{2 y}{r}$

Step 2.3.1.3

Multiply $r$ by $1$ .

$r^{2} = r + r \frac{2 y}{r}$

Step 2.3.1.4

Cancel the common factor of $r$ .

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Step 2.3.1.4.1

Cancel the common factor.

$r^{2} = r + r \frac{2 y}{r}$

Step 2.3.1.4.2

Rewrite the expression.

$r^{2} = r + 2 y$

Step 3

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} = \sqrt{x^{2} + y^{2}} + 2 y$

Step 4

Write in standard form.

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Step 4.1

Move all the expressions to the left side of the equation.

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Step 4.1.1

Subtract $\sqrt{x^{2} + y^{2}}$ from both sides of the equation.

$x^{2} + y^{2} - \sqrt{x^{2} + y^{2}} = 2 y$

Step 4.1.2

Subtract $2 y$ from both sides of the equation.

$x^{2} + y^{2} - \sqrt{x^{2} + y^{2}} - 2 y = 0$

Step 4.2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c = 0$

Step 4.3

Move $- \sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} - 2 y - \sqrt{x^{2} + y^{2}} = 0$

Step 5