Algebra Examples

${(x - 3)}^{2} + {(y + 4)}^{2} = 25$

Step 1

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

${((r \cos (θ)) - 3)}^{2} + {(y + 4)}^{2} = 25$

Step 2

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

${((r \cos (θ)) - 3)}^{2} + {((r \sin (θ)) + 4)}^{2} = 25$

Step 3

Solve for $r$ .

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Step 3.1

Subtract $25$ from both sides of the equation.

${((r \cos (θ)) - 3)}^{2} + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2

Simplify the left side of the equation.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Rewrite ${(r \cos (θ) - 3)}^{2}$ as $(r \cos (θ) - 3) (r \cos (θ) - 3)$ .

$(r \cos (θ) - 3) (r \cos (θ) - 3) + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.2

Expand $(r \cos (θ) - 3) (r \cos (θ) - 3)$ using the FOIL Method.

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Step 3.2.1.2.1

Apply the distributive property.

$r \cos (θ) (r \cos (θ) - 3) - 3 (r \cos (θ) - 3) + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.2.2

Apply the distributive property.

$r \cos (θ) (r \cos (θ)) + r \cos (θ) \cdot - 3 - 3 (r \cos (θ) - 3) + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.2.3

Apply the distributive property.

$r \cos (θ) (r \cos (θ)) + r \cos (θ) \cdot - 3 - 3 (r \cos (θ)) - 3 \cdot - 3 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.3

Simplify and combine like terms.

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Step 3.2.1.3.1

Simplify each term.

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Step 3.2.1.3.1.1

Multiply $r$ by $r$ by adding the exponents.

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Step 3.2.1.3.1.1.1

Move $r$ .

$r \cdot r \cos (θ) \cos (θ) + r \cos (θ) \cdot - 3 - 3 (r \cos (θ)) - 3 \cdot - 3 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.3.1.1.2

Multiply $r$ by $r$ .

$r^{2} \cos (θ) \cos (θ) + r \cos (θ) \cdot - 3 - 3 (r \cos (θ)) - 3 \cdot - 3 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.3.1.2

Multiply $r^{2} \cos (θ) \cos (θ)$ .

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Step 3.2.1.3.1.2.1

Raise $\cos (θ)$ to the power of $1$ .

$r^{2} (\cos (θ) \cos (θ)) + r \cos (θ) \cdot - 3 - 3 (r \cos (θ)) - 3 \cdot - 3 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.3.1.2.2

Raise $\cos (θ)$ to the power of $1$ .

$r^{2} (\cos (θ) \cos (θ)) + r \cos (θ) \cdot - 3 - 3 (r \cos (θ)) - 3 \cdot - 3 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.3.1.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r^{2} {\cos (θ)}^{1 + 1} + r \cos (θ) \cdot - 3 - 3 (r \cos (θ)) - 3 \cdot - 3 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.3.1.2.4

Add $1$ and $1$ .

$r^{2} \cos^{2} (θ) + r \cos (θ) \cdot - 3 - 3 (r \cos (θ)) - 3 \cdot - 3 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.3.1.3

Move $- 3$ to the left of $r \cos (θ)$ .

$r^{2} \cos^{2} (θ) - 3 \cdot (r \cos (θ)) - 3 (r \cos (θ)) - 3 \cdot - 3 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.3.1.4

Multiply $- 3$ by $- 3$ .

$r^{2} \cos^{2} (θ) - 3 r \cos (θ) - 3 r \cos (θ) + 9 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.3.2

Subtract $3 r \cos (θ)$ from $- 3 r \cos (θ)$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + {((r \sin (θ)) + 4)}^{2} - 25 = 0$

Step 3.2.1.4

Rewrite ${(r \sin (θ) + 4)}^{2}$ as $(r \sin (θ) + 4) (r \sin (θ) + 4)$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + (r \sin (θ) + 4) (r \sin (θ) + 4) - 25 = 0$

Step 3.2.1.5

Expand $(r \sin (θ) + 4) (r \sin (θ) + 4)$ using the FOIL Method.

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Step 3.2.1.5.1

Apply the distributive property.

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r \sin (θ) (r \sin (θ) + 4) + 4 (r \sin (θ) + 4) - 25 = 0$

Step 3.2.1.5.2

Apply the distributive property.

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r \sin (θ) (r \sin (θ)) + r \sin (θ) \cdot 4 + 4 (r \sin (θ) + 4) - 25 = 0$

Step 3.2.1.5.3

Apply the distributive property.

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r \sin (θ) (r \sin (θ)) + r \sin (θ) \cdot 4 + 4 (r \sin (θ)) + 4 \cdot 4 - 25 = 0$

Step 3.2.1.6

Simplify and combine like terms.

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Step 3.2.1.6.1

Simplify each term.

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Step 3.2.1.6.1.1

Multiply $r$ by $r$ by adding the exponents.

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Step 3.2.1.6.1.1.1

Move $r$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r \cdot r \sin (θ) \sin (θ) + r \sin (θ) \cdot 4 + 4 (r \sin (θ)) + 4 \cdot 4 - 25 = 0$

Step 3.2.1.6.1.1.2

Multiply $r$ by $r$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r^{2} \sin (θ) \sin (θ) + r \sin (θ) \cdot 4 + 4 (r \sin (θ)) + 4 \cdot 4 - 25 = 0$

Step 3.2.1.6.1.2

Multiply $r^{2} \sin (θ) \sin (θ)$ .

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Step 3.2.1.6.1.2.1

Raise $\sin (θ)$ to the power of $1$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r^{2} (\sin (θ) \sin (θ)) + r \sin (θ) \cdot 4 + 4 (r \sin (θ)) + 4 \cdot 4 - 25 = 0$

Step 3.2.1.6.1.2.2

Raise $\sin (θ)$ to the power of $1$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r^{2} (\sin (θ) \sin (θ)) + r \sin (θ) \cdot 4 + 4 (r \sin (θ)) + 4 \cdot 4 - 25 = 0$

Step 3.2.1.6.1.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r^{2} {\sin (θ)}^{1 + 1} + r \sin (θ) \cdot 4 + 4 (r \sin (θ)) + 4 \cdot 4 - 25 = 0$

Step 3.2.1.6.1.2.4

Add $1$ and $1$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r^{2} \sin^{2} (θ) + r \sin (θ) \cdot 4 + 4 (r \sin (θ)) + 4 \cdot 4 - 25 = 0$

Step 3.2.1.6.1.3

Move $4$ to the left of $r \sin (θ)$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r^{2} \sin^{2} (θ) + 4 \cdot (r \sin (θ)) + 4 (r \sin (θ)) + 4 \cdot 4 - 25 = 0$

Step 3.2.1.6.1.4

Multiply $4$ by $4$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r^{2} \sin^{2} (θ) + 4 r \sin (θ) + 4 r \sin (θ) + 16 - 25 = 0$

Step 3.2.1.6.2

Add $4 r \sin (θ)$ and $4 r \sin (θ)$ .

$r^{2} \cos^{2} (θ) - 6 r \cos (θ) + 9 + r^{2} \sin^{2} (θ) + 8 r \sin (θ) + 16 - 25 = 0$

Step 3.2.2

Simplify with factoring out.

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Step 3.2.2.1

Move $r^{2} \sin^{2} (θ)$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 6 r \cos (θ) + 9 + 8 r \sin (θ) + 16 - 25 = 0$

Step 3.2.2.2

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

$r^{2} (\cos^{2} (θ)) + r^{2} \sin^{2} (θ) - 6 r \cos (θ) + 9 + 8 r \sin (θ) + 16 - 25 = 0$

Step 3.2.2.3

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

$r^{2} (\cos^{2} (θ)) + r^{2} (\sin^{2} (θ)) - 6 r \cos (θ) + 9 + 8 r \sin (θ) + 16 - 25 = 0$

Step 3.2.2.4

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

$r^{2} (\cos^{2} (θ) + \sin^{2} (θ)) - 6 r \cos (θ) + 9 + 8 r \sin (θ) + 16 - 25 = 0$

Step 3.2.3

Rearrange terms.

$r^{2} (\sin^{2} (θ) + \cos^{2} (θ)) - 6 r \cos (θ) + 9 + 8 r \sin (θ) + 16 - 25 = 0$

Step 3.2.4

Apply pythagorean identity.

$r^{2} \cdot 1 - 6 r \cos (θ) + 9 + 8 r \sin (θ) + 16 - 25 = 0$

Step 3.2.5

Simplify by adding terms.

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Step 3.2.5.1

Multiply $r^{2}$ by $1$ .

$r^{2} - 6 r \cos (θ) + 9 + 8 r \sin (θ) + 16 - 25 = 0$

Step 3.2.5.2

Add $9$ and $16$ .

$r^{2} - 6 r \cos (θ) + 8 r \sin (θ) + 25 - 25 = 0$

Step 3.2.5.3

Combine the opposite terms in $r^{2} - 6 r \cos (θ) + 8 r \sin (θ) + 25 - 25$ .

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Step 3.2.5.3.1

Subtract $25$ from $25$ .

$r^{2} - 6 r \cos (θ) + 8 r \sin (θ) + 0 = 0$

Step 3.2.5.3.2

Add $r^{2} - 6 r \cos (θ) + 8 r \sin (θ)$ and $0$ .

$r^{2} - 6 r \cos (θ) + 8 r \sin (θ) = 0$

Step 3.3

Factor $r$ out of $r^{2} - 6 r \cos (θ) + 8 r \sin (θ)$ .

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Step 3.3.1

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

$r \cdot r - 6 r \cos (θ) + 8 r \sin (θ) = 0$

Step 3.3.2

Factor $r$ out of $- 6 r \cos (θ)$ .

$r \cdot r + r (- 6 \cos (θ)) + 8 r \sin (θ) = 0$

Step 3.3.3

Factor $r$ out of $8 r \sin (θ)$ .

$r \cdot r + r (- 6 \cos (θ)) + r (8 \sin (θ)) = 0$

Step 3.3.4

Factor $r$ out of $r \cdot r + r (- 6 \cos (θ))$ .

$r (r - 6 \cos (θ)) + r (8 \sin (θ)) = 0$

Step 3.3.5

Factor $r$ out of $r (r - 6 \cos (θ)) + r (8 \sin (θ))$ .

$r (r - 6 \cos (θ) + 8 \sin (θ)) = 0$

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 $0$ .

$r = 0$

$r - 6 \cos (θ) + 8 \sin (θ) = 0$

Step 3.5

Set $r$ equal to $0$ .

$r = 0$

Step 3.6

Set $r - 6 \cos (θ) + 8 \sin (θ)$ equal to $0$ and solve for $r$ .

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Step 3.6.1

Set $r - 6 \cos (θ) + 8 \sin (θ)$ equal to $0$ .

$r - 6 \cos (θ) + 8 \sin (θ) = 0$

Step 3.6.2

Move all terms not containing $r$ to the right side of the equation.

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Step 3.6.2.1

Add $6 \cos (θ)$ to both sides of the equation.

$r + 8 \sin (θ) = 6 \cos (θ)$

Step 3.6.2.2

Subtract $8 \sin (θ)$ from both sides of the equation.

$r = 6 \cos (θ) - 8 \sin (θ)$

Step 3.7

The final solution is all the values that make $r (r - 6 \cos (θ) + 8 \sin (θ)) = 0$ true.

$r = 0$

$r = 6 \cos (θ) - 8 \sin (θ)$

$r = 0$

$r = 6 \cos (θ) - 8 \sin (θ)$