Finite Math Examples

$x^{4} - 3 x^{3} - 18 x^{2} - 18 x - 4 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$- 18 x^{2} - 18 x + x^{4} - 3 x^{3} - 4 = 0$

Step 1.2

Factor $- 18 x$ out of $- 18 x^{2} - 18 x$ .

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Step 1.2.1

Factor $- 18 x$ out of $- 18 x^{2}$ .

$- 18 x \cdot x - 18 x + x^{4} - 3 x^{3} - 4 = 0$

Step 1.2.2

Factor $- 18 x$ out of $- 18 x$ .

$- 18 x \cdot x - 18 x \cdot 1 + x^{4} - 3 x^{3} - 4 = 0$

Step 1.2.3

Factor $- 18 x$ out of $- 18 x (x) - 18 x (1)$ .

$- 18 x (x + 1) + x^{4} - 3 x^{3} - 4 = 0$

Step 1.3

Factor $x^{4} - 3 x^{3} - 4$ using the rational roots test.

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Step 1.3.1

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.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

Step 1.3.2

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

$\pm 1, \pm 4, \pm 2$

Step 1.3.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 1.3.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - 3 {(- 1)}^{3} - 4$

Step 1.3.3.2

Raise $- 1$ to the power of $4$ .

$1 - 3 {(- 1)}^{3} - 4$

Step 1.3.3.3

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

$1 - 3 \cdot - 1 - 4$

Step 1.3.3.4

Multiply $- 3$ by $- 1$ .

$1 + 3 - 4$

Step 1.3.3.5

Add $1$ and $3$ .

$4 - 4$

Step 1.3.3.6

Subtract $4$ from $4$ .

$0$

Step 1.3.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 3 x^{3} - 4}{x + 1}$

Step 1.3.5

Divide $x^{4} - 3 x^{3} - 4$ by $x + 1$ .

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Step 1.3.5.1

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

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

Step 1.3.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

Step 1.3.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

Step 1.3.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + x^{3}$

$x^{3}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

Step 1.3.5.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

Step 1.3.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

Step 1.3.5.7

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

Step 1.3.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 1.3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{3} - 4 x^{2}$

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 1.3.5.10

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

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 1.3.5.11

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

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

Step 1.3.5.12

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

Step 1.3.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

Step 1.3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} + 4 x$

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

Step 1.3.5.15

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

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

Step 1.3.5.16

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

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

Step 1.3.5.17

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$4 x$

$4$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

Step 1.3.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$4 x$

$4$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

$4 x$

$4$

Step 1.3.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 4$

$x^{3}$

$4 x^{2}$

$4 x$

$4$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

$4 x$

$4$

Step 1.3.5.20

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

$x^{3}$

$4 x^{2}$

$4 x$

$4$

$x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

$4 x$

$4$

$0$

Step 1.3.5.21

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

$x^{3} - 4 x^{2} + 4 x - 4$

Step 1.3.6

Write $x^{4} - 3 x^{3} - 4$ as a set of factors.

$- 18 x (x + 1) + (x + 1) (x^{3} - 4 x^{2} + 4 x - 4) = 0$

Step 1.4

Factor $x + 1$ out of $- 18 x (x + 1) + (x + 1) (x^{3} - 4 x^{2} + 4 x - 4)$ .

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Step 1.4.1

Factor $x + 1$ out of $- 18 x (x + 1)$ .

$(x + 1) (- 18 x) + (x + 1) (x^{3} - 4 x^{2} + 4 x - 4) = 0$

Step 1.4.2

Factor $x + 1$ out of $(x + 1) (- 18 x) + (x + 1) (x^{3} - 4 x^{2} + 4 x - 4)$ .

$(x + 1) (- 18 x + x^{3} - 4 x^{2} + 4 x - 4) = 0$

Step 1.5

Add $- 18 x$ and $4 x$ .

$(x + 1) (x^{3} - 4 x^{2} - 14 x - 4) = 0$

Step 1.6

Factor.

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Step 1.6.1

Factor $x^{3} - 4 x^{2} - 14 x - 4$ using the rational roots test.

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Step 1.6.1.1

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.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

Step 1.6.1.2

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

$\pm 1, \pm 4, \pm 2$

Step 1.6.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 1.6.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - 4 {(- 2)}^{2} - 14 \cdot - 2 - 4$

Step 1.6.1.3.2

Raise $- 2$ to the power of $3$ .

$- 8 - 4 {(- 2)}^{2} - 14 \cdot - 2 - 4$

Step 1.6.1.3.3

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

$- 8 - 4 \cdot 4 - 14 \cdot - 2 - 4$

Step 1.6.1.3.4

Multiply $- 4$ by $4$ .

$- 8 - 16 - 14 \cdot - 2 - 4$

Step 1.6.1.3.5

Subtract $16$ from $- 8$ .

$- 24 - 14 \cdot - 2 - 4$

Step 1.6.1.3.6

Multiply $- 14$ by $- 2$ .

$- 24 + 28 - 4$

Step 1.6.1.3.7

Add $- 24$ and $28$ .

$4 - 4$

Step 1.6.1.3.8

Subtract $4$ from $4$ .

$0$

Step 1.6.1.4

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 4 x^{2} - 14 x - 4}{x + 2}$

Step 1.6.1.5

Divide $x^{3} - 4 x^{2} - 14 x - 4$ by $x + 2$ .

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Step 1.6.1.5.1

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

$x$

$2$

$x^{3}$

$4 x^{2}$

$14 x$

$4$

Step 1.6.1.5.2

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$

Step 1.6.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			+	$x^{3}$	+	$2 x^{2}$

Step 1.6.1.5.4

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$

Step 1.6.1.5.5

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$

Step 1.6.1.5.6

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$

Step 1.6.1.5.7

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$

Step 1.6.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$
					-	$6 x^{2}$	-	$12 x$

Step 1.6.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 12 x$

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$
					+	$6 x^{2}$	+	$12 x$

Step 1.6.1.5.10

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$
					+	$6 x^{2}$	+	$12 x$
							-	$2 x$

Step 1.6.1.5.11

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$
					+	$6 x^{2}$	+	$12 x$
							-	$2 x$	-	$4$

Step 1.6.1.5.12

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

				$x^{2}$	-	$6 x$	-	$2$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$
					+	$6 x^{2}$	+	$12 x$
							-	$2 x$	-	$4$

Step 1.6.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	-	$2$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$
					+	$6 x^{2}$	+	$12 x$
							-	$2 x$	-	$4$
							-	$2 x$	-	$4$

Step 1.6.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x - 4$

				$x^{2}$	-	$6 x$	-	$2$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$
					+	$6 x^{2}$	+	$12 x$
							-	$2 x$	-	$4$
							+	$2 x$	+	$4$

Step 1.6.1.5.15

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

				$x^{2}$	-	$6 x$	-	$2$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$14 x$	-	$4$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$14 x$
					+	$6 x^{2}$	+	$12 x$
							-	$2 x$	-	$4$
							+	$2 x$	+	$4$
										$0$

Step 1.6.1.5.16

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

$x^{2} - 6 x - 2$

Step 1.6.1.6

Write $x^{3} - 4 x^{2} - 14 x - 4$ as a set of factors.

$(x + 1) ((x + 2) (x^{2} - 6 x - 2)) = 0$

Step 1.6.2

Remove unnecessary parentheses.

$(x + 1) (x + 2) (x^{2} - 6 x - 2) = 0$

Step 2

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

$x + 1 = 0$

$x + 2 = 0$

$x^{2} - 6 x - 2 = 0$

Step 3

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 3.1

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

$x + 1 = 0$

Step 3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 4.1

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

$x + 2 = 0$

Step 4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Set $x^{2} - 6 x - 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} - 6 x - 2$ equal to $0$ .

$x^{2} - 6 x - 2 = 0$

Step 5.2

Solve $x^{2} - 6 x - 2 = 0$ for $x$ .

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Step 5.2.1

Use the quadratic formula to find the solutions.

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

Step 5.2.2

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 5.2.3

Simplify.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

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

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Step 5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.2.3.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{6 \pm \sqrt{36 + 8}}{2 \cdot 1}$

Step 5.2.3.1.3

Add $36$ and $8$ .

$x = \frac{6 \pm \sqrt{44}}{2 \cdot 1}$

Step 5.2.3.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 5.2.3.1.4.1

Factor $4$ out of $44$ .

$x = \frac{6 \pm \sqrt{4 (11)}}{2 \cdot 1}$

Step 5.2.3.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Step 5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{11}}{2 \cdot 1}$

Step 5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{11}}{2}$

Step 5.2.3.3

Simplify $\frac{6 \pm 2 \sqrt{11}}{2}$ .

$x = 3 \pm \sqrt{11}$

Step 5.2.4

The final answer is the combination of both solutions.

$x = 3 + \sqrt{11}, 3 - \sqrt{11}$

Step 6

The final solution is all the values that make $(x + 1) (x + 2) (x^{2} - 6 x - 2) = 0$ true.

$x = - 1, - 2, 3 + \sqrt{11}, 3 - \sqrt{11}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 1, - 2, 3 + \sqrt{11}, 3 - \sqrt{11}$

Decimal Form:

$x = - 1, - 2, 6.31662479 \dots, - 0.31662479 \dots$

Step 8