Algebra Examples

$f (x) = - 2 {(4 x^{2} + 2)}^{2} (x - 6)$

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 $x$ .

$0 = - 2 {(4 x^{2} + 2)}^{2} (x - 6)$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $- 2 {(4 x^{2} + 2)}^{2} (x - 6) = 0$ .

$- 2 {(4 x^{2} + 2)}^{2} (x - 6) = 0$

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 $0$ .

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

$x - 6 = 0$

Step 1.2.3

Set ${(4 x^{2} + 2)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.3.1

Set ${(4 x^{2} + 2)}^{2}$ equal to $0$ .

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

Step 1.2.3.2

Solve ${(4 x^{2} + 2)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 1.2.3.2.1

Factor the left side of the equation.

Tap for more steps...

Step 1.2.3.2.1.1

Factor $2$ out of $4 x^{2} + 2$ .

Tap for more steps...

Step 1.2.3.2.1.1.1

Factor $2$ out of $4 x^{2}$ .

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

Step 1.2.3.2.1.1.2

Factor $2$ out of $2$ .

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

Step 1.2.3.2.1.1.3

Factor $2$ out of $2 (2 x^{2}) + 2 (1)$ .

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

Step 1.2.3.2.1.2

Apply the product rule to $2 (2 x^{2} + 1)$ .

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

Step 1.2.3.2.2

Divide each term in $2^{2} {(2 x^{2} + 1)}^{2} = 0$ by $2^{2}$ and simplify.

Tap for more steps...

Step 1.2.3.2.2.1

Divide each term in $2^{2} {(2 x^{2} + 1)}^{2} = 0$ by $2^{2}$ .

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

Step 1.2.3.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.2.2.1

Cancel the common factor of $2^{2}$ .

Tap for more steps...

Step 1.2.3.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.2.2.1.2

Divide ${(2 x^{2} + 1)}^{2}$ by $1$ .

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

Step 1.2.3.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.2.3.1

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

${(2 x^{2} + 1)}^{2} = \frac{0}{4}$

Step 1.2.3.2.2.3.2

Divide $0$ by $4$ .

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

Step 1.2.3.2.3

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

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

Step 1.2.3.2.4

Solve for $x$ .

Tap for more steps...

Step 1.2.3.2.4.1

Subtract $1$ from both sides of the equation.

$2 x^{2} = - 1$

Step 1.2.3.2.4.2

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

Tap for more steps...

Step 1.2.3.2.4.2.1

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

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

Step 1.2.3.2.4.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.4.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.4.2.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.4.2.2.1.2

Divide $x^{2}$ by $1$ .

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

Step 1.2.3.2.4.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.4.2.3.1

Move the negative in front of the fraction.

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

Step 1.2.3.2.4.3

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

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

Step 1.2.3.2.4.4

Simplify $\pm \sqrt{- \frac{1}{2}}$ .

Tap for more steps...

Step 1.2.3.2.4.4.1

Rewrite $- \frac{1}{2}$ as $i^{2} \frac{1^{2}}{2}$ .

Tap for more steps...

Step 1.2.3.2.4.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1}{2}}$

Step 1.2.3.2.4.4.1.2

Rewrite $1$ as $1^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1^{2}}{2}}$

Step 1.2.3.2.4.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{2}}$

Step 1.2.3.2.4.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{2}}$

Step 1.2.3.2.4.4.4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{2}}$

Step 1.2.3.2.4.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{2}}$

Step 1.2.3.2.4.4.6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.2.3.2.4.4.7

Combine and simplify the denominator.

Tap for more steps...

Step 1.2.3.2.4.4.7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.2.3.2.4.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.3.2.4.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.3.2.4.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.3.2.4.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.3.2.4.4.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 1.2.3.2.4.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.3.2.4.4.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm i \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.3.2.4.4.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.3.2.4.4.7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.4.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.3.2.4.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{2}}{2^{1}}$

Step 1.2.3.2.4.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2}}{2}$

Step 1.2.3.2.4.4.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$x = \pm \frac{i \sqrt{2}}{2}$

Step 1.2.3.2.4.5

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

Tap for more steps...

Step 1.2.3.2.4.5.1

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

$x = \frac{i \sqrt{2}}{2}$

Step 1.2.3.2.4.5.2

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

$x = - \frac{i \sqrt{2}}{2}$

Step 1.2.3.2.4.5.3

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

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

Step 1.2.4

Set $x - 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.4.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 1.2.4.2

Add $6$ to both sides of the equation.

$x = 6$

Step 1.2.5

The final solution is all the values that make $- 2 {(4 x^{2} + 2)}^{2} (x - 6) = 0$ true.

$x = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}, 6$

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 $y$ .

$y = - 2 {(4 {(0)}^{2} + 2)}^{2} ((0) - 6)$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = - 2 {(4 \cdot 0^{2} + 2)}^{2} ((0) - 6)$

Step 2.2.2

Remove parentheses.

$y = - 2 {(4 \cdot 0^{2} + 2)}^{2} (0 - 6)$

Step 2.2.3

Remove parentheses.

$y = - 2 {(4 {(0)}^{2} + 2)}^{2} ((0) - 6)$

Step 2.2.4

Simplify $- 2 {(4 {(0)}^{2} + 2)}^{2} ((0) - 6)$ .

Tap for more steps...

Step 2.2.4.1

Simplify each term.

Tap for more steps...

Step 2.2.4.1.1

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

$y = - 2 {(4 \cdot 0 + 2)}^{2} ((0) - 6)$

Step 2.2.4.1.2

Multiply $4$ by $0$ .

$y = - 2 {(0 + 2)}^{2} ((0) - 6)$

Step 2.2.4.2

Simplify the expression.

Tap for more steps...

Step 2.2.4.2.1

Add $0$ and $2$ .

$y = - 2 \cdot 2^{2} ((0) - 6)$

Step 2.2.4.2.2

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

$y = - 2 \cdot 4 ((0) - 6)$

Step 2.2.4.2.3

Multiply $- 2$ by $4$ .

$y = - 8 ((0) - 6)$

Step 2.2.4.2.4

Subtract $6$ from $0$ .

$y = - 8 \cdot - 6$

Step 2.2.4.2.5

Multiply $- 8$ by $- 6$ .

$y = 48$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 48)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 48)$

Step 4