Precalculus Examples

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

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

${(x - 2)}^{2} + {((0) + 3)}^{2} = 9$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Subtract ${((0) + 3)}^{2}$ from both sides of the equation.

${(x - 2)}^{2} = 9 - {((0) + 3)}^{2}$

Step 1.2.2

Add $0$ and $3$ .

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

Step 1.2.3

Raise $3$ to the power of $2$ .

${(x - 2)}^{2} = 9 - 1 \cdot 9$

Step 1.2.4

Multiply $- 1$ by $9$ .

${(x - 2)}^{2} = 9 - 9$

Step 1.2.5

Subtract $9$ from $9$ .

${(x - 2)}^{2} = 0$

Step 1.2.6

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

$x - 2 = \pm \sqrt{0}$

Step 1.2.7

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 1.2.7.1

Rewrite $0$ as $0^{2}$ .

$x - 2 = \pm \sqrt{0^{2}}$

Step 1.2.7.2

Pull terms out from under the radical, assuming positive real numbers.

$x - 2 = \pm 0$

Step 1.2.7.3

Plus or minus $0$ is $0$ .

$x - 2 = 0$

Step 1.2.8

Add $2$ to both sides of the equation.

$x = 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

${((0) - 2)}^{2} + {(y + 3)}^{2} = 9$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Subtract ${((0) - 2)}^{2}$ from both sides of the equation.

${(y + 3)}^{2} = 9 - {((0) - 2)}^{2}$

Step 2.2.2

Subtract $2$ from $0$ .

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

Step 2.2.3

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

${(y + 3)}^{2} = 9 - 1 \cdot 4$

Step 2.2.4

Multiply $- 1$ by $4$ .

${(y + 3)}^{2} = 9 - 4$

Step 2.2.5

Subtract $4$ from $9$ .

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

Step 2.2.6

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

$y + 3 = \pm \sqrt{5}$

Step 2.2.7

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

Tap for more steps...

Step 2.2.7.1

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

$y + 3 = \sqrt{5}$

Step 2.2.7.2

Subtract $3$ from both sides of the equation.

$y = \sqrt{5} - 3$

Step 2.2.7.3

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

$y + 3 = - \sqrt{5}$

Step 2.2.7.4

Subtract $3$ from both sides of the equation.

$y = - \sqrt{5} - 3$

Step 2.2.7.5

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

$y = \sqrt{5} - 3, - \sqrt{5} - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \sqrt{5} - 3), (0, - \sqrt{5} - 3)$

Step 3

List the intersections.

x-intercept(s): $(2, 0)$

y-intercept(s): $(0, \sqrt{5} - 3), (0, - \sqrt{5} - 3)$

Step 4