Trigonometry Examples

${(x - 5)}^{2} + {(y - 3)}^{2} = 9$

Step 1

Subtract ${(y - 3)}^{2}$ from both sides of the equation.

${(x - 5)}^{2} = 9 - {(y - 3)}^{2}$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 5 = \pm \sqrt{9 - {(y - 3)}^{2}}$

Step 3

Simplify $\pm \sqrt{9 - {(y - 3)}^{2}}$ .

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$x - 5 = \pm \sqrt{3^{2} - {(y - 3)}^{2}}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = y - 3$ .

$x - 5 = \pm \sqrt{(3 + y - 3) (3 - (y - 3))}$

Step 3.3

Simplify.

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Step 3.3.1

Subtract $3$ from $3$ .

$x - 5 = \pm \sqrt{(y + 0) (3 - (y - 3))}$

Step 3.3.2

Add $y$ and $0$ .

$x - 5 = \pm \sqrt{y (3 - (y - 3))}$

Step 3.3.3

Apply the distributive property.

$x - 5 = \pm \sqrt{y (3 - y - - 3)}$

Step 3.3.4

Multiply $- 1$ by $- 3$ .

$x - 5 = \pm \sqrt{y (3 - y + 3)}$

Step 3.3.5

Add $3$ and $3$ .

$x - 5 = \pm \sqrt{y (- y + 6)}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 5 = \sqrt{y (- y + 6)}$

Step 4.2

Add $5$ to both sides of the equation.

$x = \sqrt{y (- y + 6)} + 5$

Step 4.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - 5 = - \sqrt{y (- y + 6)}$

Step 4.4

Add $5$ to both sides of the equation.

$x = - \sqrt{y (- y + 6)} + 5$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{y (- y + 6)} + 5$

$x = - \sqrt{y (- y + 6)} + 5$

$x = \sqrt{y (- y + 6)} + 5$

$x = - \sqrt{y (- y + 6)} + 5$