Algebra Examples

$f (x) = - 4 {(2 x + 1)}^{2} {(x + 2)}^{2} (x^{2} + 9) (x^{2} - 5)$

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 $x$ .

$0 = - 4 {(2 x + 1)}^{2} {(x + 2)}^{2} (x^{2} + 9) (x^{2} - 5)$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $- 4 {(2 x + 1)}^{2} {(x + 2)}^{2} (x^{2} + 9) (x^{2} - 5) = 0$ .

$- 4 {(2 x + 1)}^{2} {(x + 2)}^{2} (x^{2} + 9) (x^{2} - 5) = 0$

Step 1.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(2 x + 1)}^{2} = 0$

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

$x^{2} + 9 = 0$

$x^{2} - 5 = 0$

Step 1.2.3

Set ${(2 x + 1)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.3.1

Set ${(2 x + 1)}^{2}$ equal to $0$ .

${(2 x + 1)}^{2} = 0$

Step 1.2.3.2

Solve ${(2 x + 1)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 1.2.3.2.1

Set the $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 1.2.3.2.2

Solve for $x$ .

Tap for more steps...

Step 1.2.3.2.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 1.2.3.2.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

Tap for more steps...

Step 1.2.3.2.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 1.2.3.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.3.2.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.2.2.3.1

Move the negative in front of the fraction.

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

Step 1.2.4

Set ${(x + 2)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.4.1

Set ${(x + 2)}^{2}$ equal to $0$ .

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

Step 1.2.4.2

Solve ${(x + 2)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 1.2.4.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 1.2.4.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.5

Set $x^{2} + 9$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Step 1.2.5.2

Solve $x^{2} + 9 = 0$ for $x$ .

Tap for more steps...

Step 1.2.5.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 1.2.5.2.2

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

$x = \pm \sqrt{- 9}$

Step 1.2.5.2.3

Simplify $\pm \sqrt{- 9}$ .

Tap for more steps...

Step 1.2.5.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 1.2.5.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 1.2.5.2.3.3

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

$x = \pm i \cdot \sqrt{9}$

Step 1.2.5.2.3.4

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

$x = \pm i \cdot \sqrt{3^{2}}$

Step 1.2.5.2.3.5

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

$x = \pm i \cdot 3$

Step 1.2.5.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 1.2.5.2.4

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

Tap for more steps...

Step 1.2.5.2.4.1

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

$x = 3 i$

Step 1.2.5.2.4.2

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

$x = - 3 i$

Step 1.2.5.2.4.3

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

$x = 3 i, - 3 i$

Step 1.2.6

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.6.1

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Step 1.2.6.2

Solve $x^{2} - 5 = 0$ for $x$ .

Tap for more steps...

Step 1.2.6.2.1

Add $5$ to both sides of the equation.

$x^{2} = 5$

Step 1.2.6.2.2

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

$x = \pm \sqrt{5}$

Step 1.2.6.2.3

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

Tap for more steps...

Step 1.2.6.2.3.1

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

$x = \sqrt{5}$

Step 1.2.6.2.3.2

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

$x = - \sqrt{5}$

Step 1.2.6.2.3.3

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

$x = \sqrt{5}, - \sqrt{5}$

Step 1.2.7

The final solution is all the values that make $- 4 {(2 x + 1)}^{2} {(x + 2)}^{2} (x^{2} + 9) (x^{2} - 5) = 0$ true.

$x = - \frac{1}{2}, - 2, 3 i, - 3 i, \sqrt{5}, - \sqrt{5}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{2}, 0), (- 2, 0), (\sqrt{5}, 0), (- \sqrt{5}, 0)$

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 $y$ .

$y = - 4 {(2 (0) + 1)}^{2} {((0) + 2)}^{2} ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = - 4 {(2 (0) + 1)}^{2} {(0 + 2)}^{2} ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.2

Remove parentheses.

$y = - 4 {(2 (0) + 1)}^{2} {(0 + 2)}^{2} (0^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.3

Remove parentheses.

$y = - 4 {(2 (0) + 1)}^{2} {(0 + 2)}^{2} (0^{2} + 9) (0^{2} - 5)$

Step 2.2.4

Remove parentheses.

$y = - 4 {(2 (0) + 1)}^{2} {((0) + 2)}^{2} ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.5

Simplify $- 4 {(2 (0) + 1)}^{2} {((0) + 2)}^{2} ({(0)}^{2} + 9) ({(0)}^{2} - 5)$ .

Tap for more steps...

Step 2.2.5.1

Multiply $2$ by $0$ .

$y = - 4 {(0 + 1)}^{2} {((0) + 2)}^{2} ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.5.2

Add $0$ and $1$ .

$y = - 4 \cdot 1^{2} {((0) + 2)}^{2} ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.5.3

One to any power is one.

$y = - 4 \cdot 1 {((0) + 2)}^{2} ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.5.4

Multiply $- 4$ by $1$ .

$y = - 4 {((0) + 2)}^{2} ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.5.5

Add $0$ and $2$ .

$y = - 4 \cdot 2^{2} ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.5.6

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

$y = - 4 \cdot 4 ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.5.7

Multiply $- 4$ by $4$ .

$y = - 16 ({(0)}^{2} + 9) ({(0)}^{2} - 5)$

Step 2.2.5.8

Raising $0$ to any positive power yields $0$ .

$y = - 16 (0 + 9) ({(0)}^{2} - 5)$

Step 2.2.5.9

Add $0$ and $9$ .

$y = - 16 \cdot 9 ({(0)}^{2} - 5)$

Step 2.2.5.10

Multiply $- 16$ by $9$ .

$y = - 144 ({(0)}^{2} - 5)$

Step 2.2.5.11

Raising $0$ to any positive power yields $0$ .

$y = - 144 (0 - 5)$

Step 2.2.5.12

Subtract $5$ from $0$ .

$y = - 144 \cdot - 5$

Step 2.2.5.13

Multiply $- 144$ by $- 5$ .

$y = 720$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 720)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{1}{2}, 0), (- 2, 0), (\sqrt{5}, 0), (- \sqrt{5}, 0)$

y-intercept(s): $(0, 720)$

Step 4