Algebra Examples

$x^{2} + y^{2} = a^{2}$

Step 1

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

${(r \cos (θ))}^{2} + y^{2} = a^{2}$

Step 2

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

${(r \cos (θ))}^{2} + {(r \sin (θ))}^{2} = a^{2}$

Step 3

Solve for $r$ .

Tap for more steps...

Step 3.1

Subtract $a^{2}$ from both sides of the equation.

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

Step 3.2

Simplify the left side of the equation.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

Simplify with factoring out.

Tap for more steps...

Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.2.3

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

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

Step 3.2.3

Rearrange terms.

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

Step 3.2.4

Apply pythagorean identity.

$r^{2} \cdot 1 - a^{2} = 0$

Step 3.2.5

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

$r^{2} - a^{2} = 0$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r$ and $b = a$ .

$(r + a) (r - a) = 0$

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 $0$ .

$r + a = 0$

$r - a = 0$

Step 3.5

Set $r + a$ equal to $0$ and solve for $r$ .

Tap for more steps...

Step 3.5.1

Set $r + a$ equal to $0$ .

$r + a = 0$

Step 3.5.2

Subtract $a$ from both sides of the equation.

$r = - a$

Step 3.6

Set $r - a$ equal to $0$ and solve for $r$ .

Tap for more steps...

Step 3.6.1

Set $r - a$ equal to $0$ .

$r - a = 0$

Step 3.6.2

Add $a$ to both sides of the equation.

$r = a$

Step 3.7

The final solution is all the values that make $(r + a) (r - a) = 0$ true.

$r = - a$

$r = a$

$r = - a$

$r = a$