Calculus Examples

$X^{3} - 34 X - 12 = 0$

Step 1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

$q = \pm 1$

Step 2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(6)}^{3} - 34 \cdot 6 - 12$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $X = 6$ is a root of the polynomial.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Raise $6$ to the power of $3$ .

$216 - 34 \cdot 6 - 12$

Step 4.1.2

Multiply $- 34$ by $6$ .

$216 - 204 - 12$

Step 4.2

Simplify by subtracting numbers.

Tap for more steps...

Step 4.2.1

Subtract $204$ from $216$ .

$12 - 12$

Step 4.2.2

Subtract $12$ from $12$ .

$0$

Step 5

Since $6$ is a known root, divide the polynomial by $X - 6$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{X^{3} - 34 X - 12}{X - 6}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Tap for more steps...

Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$6$	$1$	$0$	$- 34$	$- 12$

Step 6.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$6$	$1$	$0$	$- 34$	$- 12$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(6)$ and place the result of $(6)$ under the next term in the dividend $(0)$ .

$6$	$1$	$0$	$- 34$	$- 12$
		$6$
	$1$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$6$	$1$	$0$	$- 34$	$- 12$
		$6$
	$1$	$6$

Step 6.5

Multiply the newest entry in the result $(6)$ by the divisor $(6)$ and place the result of $(36)$ under the next term in the dividend $(- 34)$ .

$6$	$1$	$0$	$- 34$	$- 12$
		$6$	$36$
	$1$	$6$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$6$	$1$	$0$	$- 34$	$- 12$
		$6$	$36$
	$1$	$6$	$2$

Step 6.7

Multiply the newest entry in the result $(2)$ by the divisor $(6)$ and place the result of $(12)$ under the next term in the dividend $(- 12)$ .

$6$	$1$	$0$	$- 34$	$- 12$
		$6$	$36$	$12$
	$1$	$6$	$2$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$6$	$1$	$0$	$- 34$	$- 12$
		$6$	$36$	$12$
	$1$	$6$	$2$	$0$

Step 6.9

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 X^{2} + 6 X + 2$

Step 6.10

Simplify the quotient polynomial.

$X^{2} + 6 X + 2$

Step 7

Solve the equation to find any remaining roots.

Tap for more steps...

Step 7.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.2

Substitute the values $a = 1$ , $b = 6$ , and $c = 2$ into the quadratic formula and solve for $X$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 7.3

Simplify.

Tap for more steps...

Step 7.3.1

Simplify the numerator.

Tap for more steps...

Step 7.3.1.1

Raise $6$ to the power of $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 7.3.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 7.3.1.2.1

Multiply $- 4$ by $1$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2}}{2 \cdot 1}$

Step 7.3.1.2.2

Multiply $- 4$ by $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 8}}{2 \cdot 1}$

Step 7.3.1.3

Subtract $8$ from $36$ .

$X = \frac{- 6 \pm \sqrt{28}}{2 \cdot 1}$

Step 7.3.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

Tap for more steps...

Step 7.3.1.4.1

Factor $4$ out of $28$ .

$X = \frac{- 6 \pm \sqrt{4 (7)}}{2 \cdot 1}$

Step 7.3.1.4.2

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

$X = \frac{- 6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Step 7.3.1.5

Pull terms out from under the radical.

$X = \frac{- 6 \pm 2 \sqrt{7}}{2 \cdot 1}$

Step 7.3.2

Multiply $2$ by $1$ .

$X = \frac{- 6 \pm 2 \sqrt{7}}{2}$

Step 7.3.3

Simplify $\frac{- 6 \pm 2 \sqrt{7}}{2}$ .

$X = - 3 \pm \sqrt{7}$

Step 7.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 7.4.1

Simplify the numerator.

Tap for more steps...

Step 7.4.1.1

Raise $6$ to the power of $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 7.4.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 7.4.1.2.1

Multiply $- 4$ by $1$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2}}{2 \cdot 1}$

Step 7.4.1.2.2

Multiply $- 4$ by $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 8}}{2 \cdot 1}$

Step 7.4.1.3

Subtract $8$ from $36$ .

$X = \frac{- 6 \pm \sqrt{28}}{2 \cdot 1}$

Step 7.4.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

Tap for more steps...

Step 7.4.1.4.1

Factor $4$ out of $28$ .

$X = \frac{- 6 \pm \sqrt{4 (7)}}{2 \cdot 1}$

Step 7.4.1.4.2

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

$X = \frac{- 6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Step 7.4.1.5

Pull terms out from under the radical.

$X = \frac{- 6 \pm 2 \sqrt{7}}{2 \cdot 1}$

Step 7.4.2

Multiply $2$ by $1$ .

$X = \frac{- 6 \pm 2 \sqrt{7}}{2}$

Step 7.4.3

Simplify $\frac{- 6 \pm 2 \sqrt{7}}{2}$ .

$X = - 3 \pm \sqrt{7}$

Step 7.4.4

Change the $\pm$ to $+$ .

$X = - 3 + \sqrt{7}$

Step 7.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 7.5.1

Simplify the numerator.

Tap for more steps...

Step 7.5.1.1

Raise $6$ to the power of $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 7.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 7.5.1.2.1

Multiply $- 4$ by $1$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2}}{2 \cdot 1}$

Step 7.5.1.2.2

Multiply $- 4$ by $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 8}}{2 \cdot 1}$

Step 7.5.1.3

Subtract $8$ from $36$ .

$X = \frac{- 6 \pm \sqrt{28}}{2 \cdot 1}$

Step 7.5.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

Tap for more steps...

Step 7.5.1.4.1

Factor $4$ out of $28$ .

$X = \frac{- 6 \pm \sqrt{4 (7)}}{2 \cdot 1}$

Step 7.5.1.4.2

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

$X = \frac{- 6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Step 7.5.1.5

Pull terms out from under the radical.

$X = \frac{- 6 \pm 2 \sqrt{7}}{2 \cdot 1}$

Step 7.5.2

Multiply $2$ by $1$ .

$X = \frac{- 6 \pm 2 \sqrt{7}}{2}$

Step 7.5.3

Simplify $\frac{- 6 \pm 2 \sqrt{7}}{2}$ .

$X = - 3 \pm \sqrt{7}$

Step 7.5.4

Change the $\pm$ to $-$ .

$X = - 3 - \sqrt{7}$

Step 7.6

The final answer is the combination of both solutions.

$X = - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Step 8

The polynomial can be written as a set of linear factors.

$(X - 6) (X + 3 - \sqrt{7}) (X + 3 + \sqrt{7})$

Step 9

These are the roots (zeros) of the polynomial $X^{3} - 34 X - 12$ .

$X = 6, - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Step 10

Factor $X^{3} - 34 X - 12$ using the rational roots test.

Tap for more steps...

Step 10.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1$

Step 10.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Step 10.3

Substitute $6$ and simplify the expression. In this case, the expression is equal to $0$ so $6$ is a root of the polynomial.

Tap for more steps...

Step 10.3.1

Substitute $6$ into the polynomial.

$6^{3} - 34 \cdot 6 - 12$

Step 10.3.2

Raise $6$ to the power of $3$ .

$216 - 34 \cdot 6 - 12$

Step 10.3.3

Multiply $- 34$ by $6$ .

$216 - 204 - 12$

Step 10.3.4

Subtract $204$ from $216$ .

$12 - 12$

Step 10.3.5

Subtract $12$ from $12$ .

$0$

Step 10.4

Since $6$ is a known root, divide the polynomial by $X - 6$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{X^{3} - 34 X - 12}{X - 6}$

Step 10.5

Divide $X^{3} - 34 X - 12$ by $X - 6$ .

Tap for more steps...

Step 10.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$X$

$6$

$X^{3}$

$0 X^{2}$

$34 X$

$12$

Step 10.5.2

Divide the highest order term in the dividend $X^{3}$ by the highest order term in divisor $X$ .

					$X^{2}$
	$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$

Step 10.5.3

Multiply the new quotient term by the divisor.

				$X^{2}$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			+	$X^{3}$	-	$6 X^{2}$

Step 10.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $X^{3} - 6 X^{2}$

				$X^{2}$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$

Step 10.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$X^{2}$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$

Step 10.5.6

Pull the next terms from the original dividend down into the current dividend.

				$X^{2}$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$

Step 10.5.7

Divide the highest order term in the dividend $6 X^{2}$ by the highest order term in divisor $X$ .

				$X^{2}$	+	$6 X$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$

Step 10.5.8

Multiply the new quotient term by the divisor.

				$X^{2}$	+	$6 X$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$
					+	$6 X^{2}$	-	$36 X$

Step 10.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 X^{2} - 36 X$

				$X^{2}$	+	$6 X$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$
					-	$6 X^{2}$	+	$36 X$

Step 10.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$X^{2}$	+	$6 X$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$
					-	$6 X^{2}$	+	$36 X$
							+	$2 X$

Step 10.5.11

Pull the next terms from the original dividend down into the current dividend.

				$X^{2}$	+	$6 X$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$
					-	$6 X^{2}$	+	$36 X$
							+	$2 X$	-	$12$

Step 10.5.12

Divide the highest order term in the dividend $2 X$ by the highest order term in divisor $X$ .

				$X^{2}$	+	$6 X$	+	$2$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$
					-	$6 X^{2}$	+	$36 X$
							+	$2 X$	-	$12$

Step 10.5.13

Multiply the new quotient term by the divisor.

				$X^{2}$	+	$6 X$	+	$2$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$
					-	$6 X^{2}$	+	$36 X$
							+	$2 X$	-	$12$
							+	$2 X$	-	$12$

Step 10.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $2 X - 12$

				$X^{2}$	+	$6 X$	+	$2$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$
					-	$6 X^{2}$	+	$36 X$
							+	$2 X$	-	$12$
							-	$2 X$	+	$12$

Step 10.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$X^{2}$	+	$6 X$	+	$2$
$X$	-	$6$		$X^{3}$	+	$0 X^{2}$	-	$34 X$	-	$12$
			-	$X^{3}$	+	$6 X^{2}$
					+	$6 X^{2}$	-	$34 X$
					-	$6 X^{2}$	+	$36 X$
							+	$2 X$	-	$12$
							-	$2 X$	+	$12$
										$0$

Step 10.5.16

Since the remander is $0$ , the final answer is the quotient.

$X^{2} + 6 X + 2$

Step 10.6

Write $X^{3} - 34 X - 12$ as a set of factors.

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

Step 11

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

$X - 6 = 0$

$X^{2} + 6 X + 2 = 0$

Step 12

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

Tap for more steps...

Step 12.1

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

$X - 6 = 0$

Step 12.2

Add $6$ to both sides of the equation.

$X = 6$

Step 13

Set $X^{2} + 6 X + 2$ equal to $0$ and solve for $X$ .

Tap for more steps...

Step 13.1

Set $X^{2} + 6 X + 2$ equal to $0$ .

$X^{2} + 6 X + 2 = 0$

Step 13.2

Solve $X^{2} + 6 X + 2 = 0$ for $X$ .

Tap for more steps...

Step 13.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 13.2.2

Substitute the values $a = 1$ , $b = 6$ , and $c = 2$ into the quadratic formula and solve for $X$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 13.2.3

Simplify.

Tap for more steps...

Step 13.2.3.1

Simplify the numerator.

Tap for more steps...

Step 13.2.3.1.1

Raise $6$ to the power of $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 13.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 13.2.3.1.2.1

Multiply $- 4$ by $1$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2}}{2 \cdot 1}$

Step 13.2.3.1.2.2

Multiply $- 4$ by $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 8}}{2 \cdot 1}$

Step 13.2.3.1.3

Subtract $8$ from $36$ .

$X = \frac{- 6 \pm \sqrt{28}}{2 \cdot 1}$

Step 13.2.3.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

Tap for more steps...

Step 13.2.3.1.4.1

Factor $4$ out of $28$ .

$X = \frac{- 6 \pm \sqrt{4 (7)}}{2 \cdot 1}$

Step 13.2.3.1.4.2

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

$X = \frac{- 6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Step 13.2.3.1.5

Pull terms out from under the radical.

$X = \frac{- 6 \pm 2 \sqrt{7}}{2 \cdot 1}$

Step 13.2.3.2

Multiply $2$ by $1$ .

$X = \frac{- 6 \pm 2 \sqrt{7}}{2}$

Step 13.2.3.3

Simplify $\frac{- 6 \pm 2 \sqrt{7}}{2}$ .

$X = - 3 \pm \sqrt{7}$

Step 13.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 13.2.4.1

Simplify the numerator.

Tap for more steps...

Step 13.2.4.1.1

Raise $6$ to the power of $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 13.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 13.2.4.1.2.1

Multiply $- 4$ by $1$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2}}{2 \cdot 1}$

Step 13.2.4.1.2.2

Multiply $- 4$ by $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 8}}{2 \cdot 1}$

Step 13.2.4.1.3

Subtract $8$ from $36$ .

$X = \frac{- 6 \pm \sqrt{28}}{2 \cdot 1}$

Step 13.2.4.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

Tap for more steps...

Step 13.2.4.1.4.1

Factor $4$ out of $28$ .

$X = \frac{- 6 \pm \sqrt{4 (7)}}{2 \cdot 1}$

Step 13.2.4.1.4.2

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

$X = \frac{- 6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Step 13.2.4.1.5

Pull terms out from under the radical.

$X = \frac{- 6 \pm 2 \sqrt{7}}{2 \cdot 1}$

Step 13.2.4.2

Multiply $2$ by $1$ .

$X = \frac{- 6 \pm 2 \sqrt{7}}{2}$

Step 13.2.4.3

Simplify $\frac{- 6 \pm 2 \sqrt{7}}{2}$ .

$X = - 3 \pm \sqrt{7}$

Step 13.2.4.4

Change the $\pm$ to $+$ .

$X = - 3 + \sqrt{7}$

Step 13.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 13.2.5.1

Simplify the numerator.

Tap for more steps...

Step 13.2.5.1.1

Raise $6$ to the power of $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 13.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 13.2.5.1.2.1

Multiply $- 4$ by $1$ .

$X = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2}}{2 \cdot 1}$

Step 13.2.5.1.2.2

Multiply $- 4$ by $2$ .

$X = \frac{- 6 \pm \sqrt{36 - 8}}{2 \cdot 1}$

Step 13.2.5.1.3

Subtract $8$ from $36$ .

$X = \frac{- 6 \pm \sqrt{28}}{2 \cdot 1}$

Step 13.2.5.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

Tap for more steps...

Step 13.2.5.1.4.1

Factor $4$ out of $28$ .

$X = \frac{- 6 \pm \sqrt{4 (7)}}{2 \cdot 1}$

Step 13.2.5.1.4.2

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

$X = \frac{- 6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Step 13.2.5.1.5

Pull terms out from under the radical.

$X = \frac{- 6 \pm 2 \sqrt{7}}{2 \cdot 1}$

Step 13.2.5.2

Multiply $2$ by $1$ .

$X = \frac{- 6 \pm 2 \sqrt{7}}{2}$

Step 13.2.5.3

Simplify $\frac{- 6 \pm 2 \sqrt{7}}{2}$ .

$X = - 3 \pm \sqrt{7}$

Step 13.2.5.4

Change the $\pm$ to $-$ .

$X = - 3 - \sqrt{7}$

Step 13.2.6

The final answer is the combination of both solutions.

$X = - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Step 14

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

$X = 6, - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$X = 6, - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Decimal Form:

$X = 6, - 0.35424868 \dots, - 5.64575131 \dots$

Step 16