Calculus Examples

$3 x^{5} - 9 x^{4} - 28 x^{3} + 84 x^{2} + 9 x - 27 = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Regroup terms.

$3 x^{5} - 28 x^{3} + 9 x - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.2

Factor $x$ out of $3 x^{5} - 28 x^{3} + 9 x$ .

Tap for more steps...

Step 1.2.1

Factor $x$ out of $3 x^{5}$ .

$x (3 x^{4}) - 28 x^{3} + 9 x - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.2.2

Factor $x$ out of $- 28 x^{3}$ .

$x (3 x^{4}) + x (- 28 x^{2}) + 9 x - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.2.3

Factor $x$ out of $9 x$ .

$x (3 x^{4}) + x (- 28 x^{2}) + x \cdot 9 - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.2.4

Factor $x$ out of $x (3 x^{4}) + x (- 28 x^{2})$ .

$x (3 x^{4} - 28 x^{2}) + x \cdot 9 - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.2.5

Factor $x$ out of $x (3 x^{4} - 28 x^{2}) + x \cdot 9$ .

$x (3 x^{4} - 28 x^{2} + 9) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (3 {(x^{2})}^{2} - 28 x^{2} + 9) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (3 u^{2} - 28 u + 9) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.5

Factor by grouping.

Tap for more steps...

Step 1.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 9 = 27$ and whose sum is $b = - 28$ .

Tap for more steps...

Step 1.5.1.1

Factor $- 28$ out of $- 28 u$ .

$x (3 u^{2} - 28 u + 9) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.5.1.2

Rewrite $- 28$ as $- 1$ plus $- 27$

$x (3 u^{2} + (- 1 - 27) u + 9) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.5.1.3

Apply the distributive property.

$x (3 u^{2} - 1 u - 27 u + 9) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.5.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.5.2.1

Group the first two terms and the last two terms.

$x ((3 u^{2} - 1 u) - 27 u + 9) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.5.2.2

Factor out the greatest common factor (GCF) from each group.

$x (u (3 u - 1) - 9 (3 u - 1)) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.5.3

Factor the polynomial by factoring out the greatest common factor, $3 u - 1$ .

$x ((3 u - 1) (u - 9)) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.6

Replace all occurrences of $u$ with $x^{2}$ .

$x ((3 x^{2} - 1) (x^{2} - 9)) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.7

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

$x ((3 x^{2} - 1) (x^{2} - 3^{2})) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.8

Factor.

Tap for more steps...

Step 1.8.1

Factor.

Tap for more steps...

Step 1.8.1.1

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

$x ((3 x^{2} - 1) ((x + 3) (x - 3))) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.8.1.2

Remove unnecessary parentheses.

$x ((3 x^{2} - 1) (x + 3) (x - 3)) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.8.2

Remove unnecessary parentheses.

$x (3 x^{2} - 1) (x + 3) (x - 3) - 9 x^{4} + 84 x^{2} - 27 = 0$

Step 1.9

Factor $3$ out of $- 9 x^{4} + 84 x^{2} - 27$ .

Tap for more steps...

Step 1.9.1

Factor $3$ out of $- 9 x^{4}$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 x^{4}) + 84 x^{2} - 27 = 0$

Step 1.9.2

Factor $3$ out of $84 x^{2}$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 x^{4}) + 3 (28 x^{2}) - 27 = 0$

Step 1.9.3

Factor $3$ out of $- 27$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 x^{4}) + 3 (28 x^{2}) + 3 (- 9) = 0$

Step 1.9.4

Factor $3$ out of $3 (- 3 x^{4}) + 3 (28 x^{2})$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 x^{4} + 28 x^{2}) + 3 (- 9) = 0$

Step 1.9.5

Factor $3$ out of $3 (- 3 x^{4} + 28 x^{2}) + 3 (- 9)$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 x^{4} + 28 x^{2} - 9) = 0$

Step 1.10

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 {(x^{2})}^{2} + 28 x^{2} - 9) = 0$

Step 1.11

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 u^{2} + 28 u - 9) = 0$

Step 1.12

Factor by grouping.

Tap for more steps...

Step 1.12.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot - 9 = 27$ and whose sum is $b = 28$ .

Tap for more steps...

Step 1.12.1.1

Factor $28$ out of $28 u$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 u^{2} + 28 (u) - 9) = 0$

Step 1.12.1.2

Rewrite $28$ as $1$ plus $27$

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 u^{2} + (1 + 27) u - 9) = 0$

Step 1.12.1.3

Apply the distributive property.

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 u^{2} + 1 u + 27 u - 9) = 0$

Step 1.12.1.4

Multiply $u$ by $1$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 u^{2} + u + 27 u - 9) = 0$

Step 1.12.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.12.2.1

Group the first two terms and the last two terms.

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 ((- 3 u^{2} + u) + 27 u - 9) = 0$

Step 1.12.2.2

Factor out the greatest common factor (GCF) from each group.

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (u (- 3 u + 1) - 9 (- 3 u + 1)) = 0$

Step 1.12.3

Factor the polynomial by factoring out the greatest common factor, $- 3 u + 1$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 ((- 3 u + 1) (u - 9)) = 0$

Step 1.13

Replace all occurrences of $u$ with $x^{2}$ .

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 ((- 3 x^{2} + 1) (x^{2} - 9)) = 0$

Step 1.14

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

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 ((- 3 x^{2} + 1) (x^{2} - 3^{2})) = 0$

Step 1.15

Factor.

Tap for more steps...

Step 1.15.1

Factor.

Tap for more steps...

Step 1.15.1.1

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

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 ((- 3 x^{2} + 1) ((x + 3) (x - 3))) = 0$

Step 1.15.1.2

Remove unnecessary parentheses.

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 ((- 3 x^{2} + 1) (x + 3) (x - 3)) = 0$

Step 1.15.2

Remove unnecessary parentheses.

$x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 x^{2} + 1) (x + 3) (x - 3) = 0$

Step 1.16

Factor $(x + 3) (x - 3)$ out of $x (3 x^{2} - 1) (x + 3) (x - 3) + 3 (- 3 x^{2} + 1) (x + 3) (x - 3)$ .

Tap for more steps...

Step 1.16.1

Factor $(x + 3) (x - 3)$ out of $x (3 x^{2} - 1) (x + 3) (x - 3)$ .

$(x + 3) (x - 3) (x (3 x^{2} - 1)) + 3 (- 3 x^{2} + 1) (x + 3) (x - 3) = 0$

Step 1.16.2

Factor $(x + 3) (x - 3)$ out of $3 (- 3 x^{2} + 1) (x + 3) (x - 3)$ .

$(x + 3) (x - 3) (x (3 x^{2} - 1)) + (x + 3) (x - 3) (3 (- 3 x^{2} + 1)) = 0$

Step 1.16.3

Factor $(x + 3) (x - 3)$ out of $(x + 3) (x - 3) (x (3 x^{2} - 1)) + (x + 3) (x - 3) (3 (- 3 x^{2} + 1))$ .

$(x + 3) (x - 3) (x (3 x^{2} - 1) + 3 (- 3 x^{2} + 1)) = 0$

Step 1.17

Apply the distributive property.

$(x + 3) (x - 3) (x (3 x^{2}) + x \cdot - 1 + 3 (- 3 x^{2} + 1)) = 0$

Step 1.18

Rewrite using the commutative property of multiplication.

$(x + 3) (x - 3) (3 x \cdot x^{2} + x \cdot - 1 + 3 (- 3 x^{2} + 1)) = 0$

Step 1.19

Move $- 1$ to the left of $x$ .

$(x + 3) (x - 3) (3 x \cdot x^{2} - 1 \cdot x + 3 (- 3 x^{2} + 1)) = 0$

Step 1.20

Simplify each term.

Tap for more steps...

Step 1.20.1

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.20.1.1

Move $x^{2}$ .

$(x + 3) (x - 3) (3 (x^{2} x) - 1 \cdot x + 3 (- 3 x^{2} + 1)) = 0$

Step 1.20.1.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 1.20.1.2.1

Raise $x$ to the power of $1$ .

$(x + 3) (x - 3) (3 (x^{2} x) - 1 \cdot x + 3 (- 3 x^{2} + 1)) = 0$

Step 1.20.1.2.2

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

$(x + 3) (x - 3) (3 x^{2 + 1} - 1 \cdot x + 3 (- 3 x^{2} + 1)) = 0$

Step 1.20.1.3

Add $2$ and $1$ .

$(x + 3) (x - 3) (3 x^{3} - 1 \cdot x + 3 (- 3 x^{2} + 1)) = 0$

$(x + 3) (x - 3) (3 x^{3} - 1 x + 3 (- 3 x^{2} + 1)) = 0$

Step 1.20.2

Rewrite $- 1 x$ as $- x$ .

$(x + 3) (x - 3) (3 x^{3} - x + 3 (- 3 x^{2} + 1)) = 0$

Step 1.21

Apply the distributive property.

$(x + 3) (x - 3) (3 x^{3} - x + 3 (- 3 x^{2}) + 3 \cdot 1) = 0$

Step 1.22

Multiply $- 3$ by $3$ .

$(x + 3) (x - 3) (3 x^{3} - x - 9 x^{2} + 3 \cdot 1) = 0$

Step 1.23

Multiply $3$ by $1$ .

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

Step 1.24

Reorder terms.

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

Step 1.25

Factor.

Tap for more steps...

Step 1.25.1

Rewrite $3 x^{3} - 9 x^{2} - x + 3$ in a factored form.

Tap for more steps...

Step 1.25.1.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.25.1.1.1

Group the first two terms and the last two terms.

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

Step 1.25.1.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 1.25.1.2

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$(x + 3) (x - 3) ((x - 3) (3 x^{2} - 1)) = 0$

Step 1.25.2

Remove unnecessary parentheses.

$(x + 3) (x - 3) (x - 3) (3 x^{2} - 1) = 0$

Step 1.26

Combine exponents.

Tap for more steps...

Step 1.26.1

Raise $x - 3$ to the power of $1$ .

$(x + 3) ((x - 3) (x - 3)) (3 x^{2} - 1) = 0$

Step 1.26.2

Raise $x - 3$ to the power of $1$ .

$(x + 3) ((x - 3) (x - 3)) (3 x^{2} - 1) = 0$

Step 1.26.3

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

$(x + 3) {(x - 3)}^{1 + 1} (3 x^{2} - 1) = 0$

Step 1.26.4

Add $1$ and $1$ .

$(x + 3) {(x - 3)}^{2} (3 x^{2} - 1) = 0$

Step 2

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

$x + 3 = 0$

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

$3 x^{2} - 1 = 0$

Step 3

Set $x + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Step 4.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

Set $3 x^{2} - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.1

Set $3 x^{2} - 1$ equal to $0$ .

$3 x^{2} - 1 = 0$

Step 5.2

Solve $3 x^{2} - 1 = 0$ for $x$ .

Tap for more steps...

Step 5.2.1

Add $1$ to both sides of the equation.

$3 x^{2} = 1$

Step 5.2.2

Divide each term in $3 x^{2} = 1$ by $3$ and simplify.

Tap for more steps...

Step 5.2.2.1

Divide each term in $3 x^{2} = 1$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{1}{3}$

Step 5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{1}{3}$

Step 5.2.2.2.1.2

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

$x^{2} = \frac{1}{3}$

Step 5.2.3

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

$x = \pm \sqrt{\frac{1}{3}}$

Step 5.2.4

Simplify $\pm \sqrt{\frac{1}{3}}$ .

Tap for more steps...

Step 5.2.4.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 5.2.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{3}}$

Step 5.2.4.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.2.4.4

Combine and simplify the denominator.

Tap for more steps...

Step 5.2.4.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.2.4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 5.2.4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 5.2.4.4.4

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

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.2.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.2.4.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 5.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.2.4.4.6.2

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.2.4.4.6.3

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.2.4.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.2.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3}}{3^{1}}$

Step 5.2.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3}}{3}$

Step 5.2.5

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

Tap for more steps...

Step 5.2.5.1

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

$x = \frac{\sqrt{3}}{3}$

Step 5.2.5.2

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

$x = - \frac{\sqrt{3}}{3}$

Step 5.2.5.3

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

$x = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 6

The final solution is all the values that make $(x + 3) {(x - 3)}^{2} (3 x^{2} - 1) = 0$ true.

$x = - 3, 3, \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 3, 3, \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Decimal Form:

$x = - 3, 3, 0.57735026 \dots, - 0.57735026 \dots$