Trigonometry Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $- 4 (x + 2) = {((0) + 4)}^{2}$ .

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

Step 1.2.2

Add $0$ and $4$ .

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

Step 1.2.3

Divide each term in $- 4 (x + 2) = 4^{2}$ by $- 4$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $- 4 (x + 2) = 4^{2}$ by $- 4$ .

$\frac{- 4 (x + 2)}{- 4} = \frac{4^{2}}{- 4}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 4 (x + 2)}{- 4} = \frac{4^{2}}{- 4}$

Step 1.2.3.2.1.2

Divide $x + 2$ by $1$ .

$x + 2 = \frac{4^{2}}{- 4}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Cancel the common factor of $4^{2}$ and $- 4$ .

Tap for more steps...

Step 1.2.3.3.1.1

Rewrite $4$ as $- 1 (- 4)$ .

$x + 2 = \frac{{(- 1 (- 4))}^{2}}{- 4}$

Step 1.2.3.3.1.2

Apply the product rule to $- 1 (- 4)$ .

$x + 2 = \frac{{(- 1)}^{2} {(- 4)}^{2}}{- 4}$

Step 1.2.3.3.1.3

Raise $- 1$ to the power of $2$ .

$x + 2 = \frac{1 {(- 4)}^{2}}{- 4}$

Step 1.2.3.3.1.4

Multiply ${(- 4)}^{2}$ by $1$ .

$x + 2 = \frac{{(- 4)}^{2}}{- 4}$

Step 1.2.3.3.1.5

Factor $- 4$ out of ${(- 4)}^{2}$ .

$x + 2 = \frac{- 4 \cdot - 4}{- 4}$

Step 1.2.3.3.1.6

Cancel the common factors.

Tap for more steps...

Step 1.2.3.3.1.6.1

Factor $- 4$ out of $- 4$ .

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

Step 1.2.3.3.1.6.2

Cancel the common factor.

$x + 2 = \frac{- 4 \cdot - 4}{- 4 \cdot 1}$

Step 1.2.3.3.1.6.3

Rewrite the expression.

$x + 2 = \frac{- 4}{1}$

Step 1.2.3.3.1.6.4

Divide $- 4$ by $1$ .

$x + 2 = - 4$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

Subtract $2$ from both sides of the equation.

$x = - 4 - 2$

Step 1.2.4.2

Subtract $2$ from $- 4$ .

$x = - 6$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 6, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Add $0$ and $2$ .

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

Step 2.2.2

Multiply $- 4$ by $2$ .

${(y + 4)}^{2} = - 8$

Step 2.2.3

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

$y + 4 = \pm \sqrt{- 8}$

Step 2.2.4

Simplify $\pm \sqrt{- 8}$ .

Tap for more steps...

Step 2.2.4.1

Rewrite $- 8$ as $- 1 (8)$ .

$y + 4 = \pm \sqrt{- 1 (8)}$

Step 2.2.4.2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$y + 4 = \pm \sqrt{- 1} \cdot \sqrt{8}$

Step 2.2.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

$y + 4 = \pm i \cdot \sqrt{8}$

Step 2.2.4.4

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 2.2.4.4.1

Factor $4$ out of $8$ .

$y + 4 = \pm i \cdot \sqrt{4 (2)}$

Step 2.2.4.4.2

Rewrite $4$ as $2^{2}$ .

$y + 4 = \pm i \cdot \sqrt{2^{2} \cdot 2}$

Step 2.2.4.5

Pull terms out from under the radical.

$y + 4 = \pm i \cdot (2 \sqrt{2})$

Step 2.2.4.6

Move $2$ to the left of $i$ .

$y + 4 = \pm 2 i \sqrt{2}$

Step 2.2.5

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

Tap for more steps...

Step 2.2.5.1

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

$y + 4 = 2 i \sqrt{2}$

Step 2.2.5.2

Subtract $4$ from both sides of the equation.

$y = 2 i \sqrt{2} - 4$

Step 2.2.5.3

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

$y + 4 = - 2 i \sqrt{2}$

Step 2.2.5.4

Subtract $4$ from both sides of the equation.

$y = - 2 i \sqrt{2} - 4$

Step 2.2.5.5

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

$y = 2 i \sqrt{2} - 4, - 2 i \sqrt{2} - 4$

Step 2.3

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

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(- 6, 0)$

y-intercept(s): $None$

Step 4