Algebra Examples

$r = 8 \sin (θ) - 2 \cos (θ)$

Step 1

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

$r = 8 \frac{y}{r} - 2 \cos (θ)$

Step 2

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

$r = 8 \frac{y}{r} - 2 \frac{x}{r}$

Step 3

Multiply both sides by $r$ .

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Step 3.1

Multiply both sides by $r$ .

$r \cdot r = r (8 \frac{y}{r} - 2 \frac{x}{r})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $r$ by $r$ .

$r^{2} = r (8 \frac{y}{r} - 2 \frac{x}{r})$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $r (8 \frac{y}{r} - 2 \frac{x}{r})$ .

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Step 3.3.1.1

Simplify each term.

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Step 3.3.1.1.1

Combine $8$ and $\frac{y}{r}$ .

$r^{2} = r (\frac{8 y}{r} - 2 \frac{x}{r})$

Step 3.3.1.1.2

Combine $- 2$ and $\frac{x}{r}$ .

$r^{2} = r (\frac{8 y}{r} + \frac{- 2 x}{r})$

Step 3.3.1.1.3

Move the negative in front of the fraction.

$r^{2} = r (\frac{8 y}{r} - \frac{2 x}{r})$

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

$r^{2} = r \frac{8 y}{r} + r (- \frac{2 x}{r})$

Step 3.3.1.2.2

Cancel the common factor of $r$ .

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Step 3.3.1.2.2.1

Cancel the common factor.

$r^{2} = r \frac{8 y}{r} + r (- \frac{2 x}{r})$

Step 3.3.1.2.2.2

Rewrite the expression.

$r^{2} = 8 y + r (- \frac{2 x}{r})$

Step 3.3.1.2.3

Rewrite using the commutative property of multiplication.

$r^{2} = 8 y - r \frac{2 x}{r}$

Step 3.3.1.3

Simplify each term.

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Step 3.3.1.3.1

Cancel the common factor of $r$ .

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Step 3.3.1.3.1.1

Factor $r$ out of $- r$ .

$r^{2} = 8 y + r \cdot - 1 \frac{2 x}{r}$

Step 3.3.1.3.1.2

Cancel the common factor.

$r^{2} = 8 y + r \cdot - 1 \frac{2 x}{r}$

Step 3.3.1.3.1.3

Rewrite the expression.

$r^{2} = 8 y - (2 x)$

Step 3.3.1.3.2

Multiply $2$ by $- 1$ .

$r^{2} = 8 y - 2 x$

Step 4

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

$x^{2} + y^{2} = 8 y - 2 x$

Step 5

Write in standard form.

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Step 5.1

Move all terms containing variables to the left side of the equation.

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Step 5.1.1

Subtract $8 y$ from both sides of the equation.

$x^{2} + y^{2} - 8 y = - 2 x$

Step 5.1.2

Add $2 x$ to both sides of the equation.

$x^{2} + y^{2} - 8 y + 2 x = 0$

Step 5.1.3

Move $- 8 y$ .

$x^{2} + y^{2} + 2 x - 8 y = 0$

Step 5.2

Complete the square for $x^{2} + 2 x$ .

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Step 5.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = 2$

$c = 0$

Step 5.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 \cdot 1}$

Step 5.2.3.2

Cancel the common factor of $2$ .

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Step 5.2.3.2.1

Cancel the common factor.

$d = \frac{2}{2 \cdot 1}$

Step 5.2.3.2.2

Rewrite the expression.

$d = 1$

Step 5.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{2^{2}}{4 \cdot 1}$

Step 5.2.4.2

Simplify the right side.

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Step 5.2.4.2.1

Simplify each term.

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Step 5.2.4.2.1.1

Raise $2$ to the power of $2$ .

$e = 0 - \frac{4}{4 \cdot 1}$

Step 5.2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{4}{4}$

Step 5.2.4.2.1.3

Cancel the common factor of $4$ .

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Step 5.2.4.2.1.3.1

Cancel the common factor.

$e = 0 - \frac{4}{4}$

Step 5.2.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 5.2.4.2.1.4

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 5.2.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 5.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 1)}^{2} - 1$ .

${(x + 1)}^{2} - 1$

Step 5.3

Substitute ${(x + 1)}^{2} - 1$ for $x^{2} + 2 x$ in the equation $x^{2} + y^{2} + 2 x - 8 y = 0$ .

${(x + 1)}^{2} - 1 + y^{2} - 8 y = 0$

Step 5.4

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

${(x + 1)}^{2} + y^{2} - 8 y = 0 + 1$

Step 5.5

Complete the square for $y^{2} - 8 y$ .

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Step 5.5.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 8$

$c = 0$

Step 5.5.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.5.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 8}{2 \cdot 1}$

Step 5.5.3.2

Cancel the common factor of $- 8$ and $2$ .

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Step 5.5.3.2.1

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 \cdot 1}$

Step 5.5.3.2.2

Cancel the common factors.

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Step 5.5.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 4}{2 (1)}$

Step 5.5.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot 1}$

Step 5.5.3.2.2.3

Rewrite the expression.

$d = \frac{- 4}{1}$

Step 5.5.3.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

Step 5.5.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 1}$

Step 5.5.4.2

Simplify the right side.

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Step 5.5.4.2.1

Simplify each term.

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Step 5.5.4.2.1.1

Raise $- 8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 1}$

Step 5.5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{64}{4}$

Step 5.5.4.2.1.3

Divide $64$ by $4$ .

$e = 0 - 1 \cdot 16$

Step 5.5.4.2.1.4

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Step 5.5.4.2.2

Subtract $16$ from $0$ .

$e = - 16$

Step 5.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 4)}^{2} - 16$ .

${(y - 4)}^{2} - 16$

Step 5.6

Substitute ${(y - 4)}^{2} - 16$ for $y^{2} - 8 y$ in the equation $x^{2} + y^{2} + 2 x - 8 y = 0$ .

${(x + 1)}^{2} + {(y - 4)}^{2} - 16 = 0 + 1$

Step 5.7

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

${(x + 1)}^{2} + {(y - 4)}^{2} = 0 + 1 + 16$

Step 5.8

Simplify $0 + 1 + 16$ .

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Step 5.8.1

Add $0$ and $1$ .

${(x + 1)}^{2} + {(y - 4)}^{2} = 1 + 16$

Step 5.8.2

Add $1$ and $16$ .

${(x + 1)}^{2} + {(y - 4)}^{2} = 17$