Precalculus Examples

$2 x^{3} - 17 x^{2} - 11 x - 1 = 0$

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.

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}$

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.

$2 {(- \frac{1}{2})}^{3} - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - \frac{1}{2}$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

$2 ({(- 1)}^{3} {(\frac{1}{2})}^{3}) - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.1.2

Apply the product rule to $\frac{1}{2}$ .

$2 ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.2

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

$2 (- \frac{1^{3}}{2^{3}}) - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.3

One to any power is one.

$2 (- \frac{1}{2^{3}}) - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.4

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

$2 (- \frac{1}{8}) - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.5

Cancel the common factor of $2$ .

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Step 4.1.5.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$2 (\frac{- 1}{8}) - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.5.2

Factor $2$ out of $8$ .

$2 \frac{- 1}{2 (4)} - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.5.3

Cancel the common factor.

$2 \frac{- 1}{2 \cdot 4} - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.5.4

Rewrite the expression.

$\frac{- 1}{4} - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.6

Move the negative in front of the fraction.

$- \frac{1}{4} - 17 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 1$

Step 4.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.7.1

Apply the product rule to $- \frac{1}{2}$ .

$- \frac{1}{4} - 17 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) - 11 (- \frac{1}{2}) - 1$

Step 4.1.7.2

Apply the product rule to $\frac{1}{2}$ .

$- \frac{1}{4} - 17 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) - 11 (- \frac{1}{2}) - 1$

Step 4.1.8

Raise $- 1$ to the power of $2$ .

$- \frac{1}{4} - 17 (1 \frac{1^{2}}{2^{2}}) - 11 (- \frac{1}{2}) - 1$

Step 4.1.9

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

$- \frac{1}{4} - 17 \frac{1^{2}}{2^{2}} - 11 (- \frac{1}{2}) - 1$

Step 4.1.10

One to any power is one.

$- \frac{1}{4} - 17 \frac{1}{2^{2}} - 11 (- \frac{1}{2}) - 1$

Step 4.1.11

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

$- \frac{1}{4} - 17 (\frac{1}{4}) - 11 (- \frac{1}{2}) - 1$

Step 4.1.12

Combine $- 17$ and $\frac{1}{4}$ .

$- \frac{1}{4} + \frac{- 17}{4} - 11 (- \frac{1}{2}) - 1$

Step 4.1.13

Move the negative in front of the fraction.

$- \frac{1}{4} - \frac{17}{4} - 11 (- \frac{1}{2}) - 1$

Step 4.1.14

Multiply $- 11 (- \frac{1}{2})$ .

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Step 4.1.14.1

Multiply $- 1$ by $- 11$ .

$- \frac{1}{4} - \frac{17}{4} + 11 (\frac{1}{2}) - 1$

Step 4.1.14.2

Combine $11$ and $\frac{1}{2}$ .

$- \frac{1}{4} - \frac{17}{4} + \frac{11}{2} - 1$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$- 1 + \frac{- 1 - 17}{4} + \frac{11}{2}$

Step 4.2.2

Subtract $17$ from $- 1$ .

$- 1 + \frac{- 18}{4} + \frac{11}{2}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $- 1$ as a fraction with denominator $1$ .

$\frac{- 1}{1} + \frac{- 18}{4} + \frac{11}{2}$

Step 4.3.2

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

$\frac{- 1}{1} \cdot \frac{4}{4} + \frac{- 18}{4} + \frac{11}{2}$

Step 4.3.3

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

$\frac{- 1 \cdot 4}{4} + \frac{- 18}{4} + \frac{11}{2}$

Step 4.3.4

Multiply $\frac{11}{2}$ by $\frac{2}{2}$ .

$\frac{- 1 \cdot 4}{4} + \frac{- 18}{4} + \frac{11}{2} \cdot \frac{2}{2}$

Step 4.3.5

Multiply $\frac{11}{2}$ by $\frac{2}{2}$ .

$\frac{- 1 \cdot 4}{4} + \frac{- 18}{4} + \frac{11 \cdot 2}{2 \cdot 2}$

Step 4.3.6

Multiply $2$ by $2$ .

$\frac{- 1 \cdot 4}{4} + \frac{- 18}{4} + \frac{11 \cdot 2}{4}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 1 \cdot 4 - 18 + 11 \cdot 2}{4}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $- 1$ by $4$ .

$\frac{- 4 - 18 + 11 \cdot 2}{4}$

Step 4.5.2

Multiply $11$ by $2$ .

$\frac{- 4 - 18 + 22}{4}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Subtract $18$ from $- 4$ .

$\frac{- 22 + 22}{4}$

Step 4.6.2

Add $- 22$ and $22$ .

$\frac{0}{4}$

Step 4.6.3

Divide $0$ by $4$ .

$0$

Step 5

Since $- \frac{1}{2}$ is a known root, divide the polynomial by $x + \frac{1}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - 17 x^{2} - 11 x - 1}{x + \frac{1}{2}}$

Step 6

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

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Step 6.1

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

$- \frac{1}{2}$	$2$	$- 17$	$- 11$	$- 1$

Step 6.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{2}$	$2$	$- 17$	$- 11$	$- 1$

	$2$

Step 6.3

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 1)$ under the next term in the dividend $(- 17)$ .

$- \frac{1}{2}$	$2$	$- 17$	$- 11$	$- 1$
		$- 1$
	$2$

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.

$- \frac{1}{2}$	$2$	$- 17$	$- 11$	$- 1$
		$- 1$
	$2$	$- 18$

Step 6.5

Multiply the newest entry in the result $(- 18)$ by the divisor $(- \frac{1}{2})$ and place the result of $(9)$ under the next term in the dividend $(- 11)$ .

$- \frac{1}{2}$	$2$	$- 17$	$- 11$	$- 1$
		$- 1$	$9$
	$2$	$- 18$

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.

$- \frac{1}{2}$	$2$	$- 17$	$- 11$	$- 1$
		$- 1$	$9$
	$2$	$- 18$	$- 2$

Step 6.7

Multiply the newest entry in the result $(- 2)$ by the divisor $(- \frac{1}{2})$ and place the result of $(1)$ under the next term in the dividend $(- 1)$ .

$- \frac{1}{2}$	$2$	$- 17$	$- 11$	$- 1$
		$- 1$	$9$	$1$
	$2$	$- 18$	$- 2$

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.

$- \frac{1}{2}$	$2$	$- 17$	$- 11$	$- 1$
		$- 1$	$9$	$1$
	$2$	$- 18$	$- 2$	$0$

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.

$2 x^{2} + - 18 x - 2$

Step 6.10

Simplify the quotient polynomial.

$2 x^{2} - 18 x - 2$

Step 7

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

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Step 7.1

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

$(x + \frac{1}{2}) (2 (x^{2}) - 18 x - 2)$

Step 7.2

Factor $2$ out of $- 18 x$ .

$(x + \frac{1}{2}) (2 (x^{2}) + 2 (- 9 x) - 2)$

Step 7.3

Factor $2$ out of $- 2$ .

$(x + \frac{1}{2}) (2 (x^{2}) + 2 (- 9 x) + 2 (- 1))$

Step 7.4

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

$(x + \frac{1}{2}) (2 (x^{2} - 9 x) + 2 (- 1))$

Step 7.5

Factor $2$ out of $2 (x^{2} - 9 x) + 2 (- 1)$ .

$(x + \frac{1}{2}) (2 (x^{2} - 9 x - 1))$

Step 8

Factor $2 x^{3} - 17 x^{2} - 11 x - 1$ using the rational roots test.

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Step 8.1

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.

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 8.2

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

$\pm 1, \pm 0.5$

Step 8.3

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

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Step 8.3.1

Substitute $- 0.5$ into the polynomial.

$2 {(- 0.5)}^{3} - 17 {(- 0.5)}^{2} - 11 \cdot - 0.5 - 1$

Step 8.3.2

Raise $- 0.5$ to the power of $3$ .

$2 \cdot - 0.125 - 17 {(- 0.5)}^{2} - 11 \cdot - 0.5 - 1$

Step 8.3.3

Multiply $2$ by $- 0.125$ .

$- 0.25 - 17 {(- 0.5)}^{2} - 11 \cdot - 0.5 - 1$

Step 8.3.4

Raise $- 0.5$ to the power of $2$ .

$- 0.25 - 17 \cdot 0.25 - 11 \cdot - 0.5 - 1$

Step 8.3.5

Multiply $- 17$ by $0.25$ .

$- 0.25 - 4.25 - 11 \cdot - 0.5 - 1$

Step 8.3.6

Subtract $4.25$ from $- 0.25$ .

$- 4.5 - 11 \cdot - 0.5 - 1$

Step 8.3.7

Multiply $- 11$ by $- 0.5$ .

$- 4.5 + 5.5 - 1$

Step 8.3.8

Add $- 4.5$ and $5.5$ .

$1 - 1$

Step 8.3.9

Subtract $1$ from $1$ .

$0$

Step 8.4

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - 17 x^{2} - 11 x - 1}{2 x + 1}$

Step 8.5

Divide $2 x^{3} - 17 x^{2} - 11 x - 1$ by $2 x + 1$ .

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Step 8.5.1

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

$2 x$

$1$

$2 x^{3}$

$17 x^{2}$

$11 x$

$1$

Step 8.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $2 x$ .

					$x^{2}$
	$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$

Step 8.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			+	$2 x^{3}$	+	$x^{2}$

Step 8.5.4

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$

Step 8.5.5

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$

Step 8.5.6

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$

Step 8.5.7

Divide the highest order term in the dividend $- 18 x^{2}$ by the highest order term in divisor $2 x$ .

				$x^{2}$	-	$9 x$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$

Step 8.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$
					-	$18 x^{2}$	-	$9 x$

Step 8.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{2} - 9 x$

				$x^{2}$	-	$9 x$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$
					+	$18 x^{2}$	+	$9 x$

Step 8.5.10

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

				$x^{2}$	-	$9 x$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$
					+	$18 x^{2}$	+	$9 x$
							-	$2 x$

Step 8.5.11

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

				$x^{2}$	-	$9 x$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$
					+	$18 x^{2}$	+	$9 x$
							-	$2 x$	-	$1$

Step 8.5.12

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

				$x^{2}$	-	$9 x$	-	$1$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$
					+	$18 x^{2}$	+	$9 x$
							-	$2 x$	-	$1$

Step 8.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$	-	$1$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$
					+	$18 x^{2}$	+	$9 x$
							-	$2 x$	-	$1$
							-	$2 x$	-	$1$

Step 8.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x - 1$

				$x^{2}$	-	$9 x$	-	$1$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$
					+	$18 x^{2}$	+	$9 x$
							-	$2 x$	-	$1$
							+	$2 x$	+	$1$

Step 8.5.15

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

				$x^{2}$	-	$9 x$	-	$1$
$2 x$	+	$1$		$2 x^{3}$	-	$17 x^{2}$	-	$11 x$	-	$1$
			-	$2 x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	-	$11 x$
					+	$18 x^{2}$	+	$9 x$
							-	$2 x$	-	$1$
							+	$2 x$	+	$1$
										$0$

Step 8.5.16

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

$x^{2} - 9 x - 1$

Step 8.6

Write $2 x^{3} - 17 x^{2} - 11 x - 1$ as a set of factors.

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

Step 9

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

$2 x + 1 = 0$

$x^{2} - 9 x - 1 = 0$

Step 10

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 10.1

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

$2 x + 1 = 0$

Step 10.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 10.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 10.2.2

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

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Step 10.2.2.1

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

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

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $2$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

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

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 10.2.2.3

Simplify the right side.

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Step 10.2.2.3.1

Move the negative in front of the fraction.

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

Step 11

Set $x^{2} - 9 x - 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x^{2} - 9 x - 1$ equal to $0$ .

$x^{2} - 9 x - 1 = 0$

Step 11.2

Solve $x^{2} - 9 x - 1 = 0$ for $x$ .

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Step 11.2.1

Use the quadratic formula to find the solutions.

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

Step 11.2.2

Substitute the values $a = 1$ , $b = - 9$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 11.2.3

Simplify.

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Step 11.2.3.1

Simplify the numerator.

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Step 11.2.3.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 11.2.3.1.2

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

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Step 11.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot - 1}}{2 \cdot 1}$

Step 11.2.3.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{9 \pm \sqrt{81 + 4}}{2 \cdot 1}$

Step 11.2.3.1.3

Add $81$ and $4$ .

$x = \frac{9 \pm \sqrt{85}}{2 \cdot 1}$

Step 11.2.3.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{85}}{2}$

Step 11.2.4

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

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Step 11.2.4.1

Simplify the numerator.

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Step 11.2.4.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 11.2.4.1.2

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

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Step 11.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot - 1}}{2 \cdot 1}$

Step 11.2.4.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{9 \pm \sqrt{81 + 4}}{2 \cdot 1}$

Step 11.2.4.1.3

Add $81$ and $4$ .

$x = \frac{9 \pm \sqrt{85}}{2 \cdot 1}$

Step 11.2.4.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{85}}{2}$

Step 11.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{9 + \sqrt{85}}{2}$

Step 11.2.5

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

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Step 11.2.5.1

Simplify the numerator.

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Step 11.2.5.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 11.2.5.1.2

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

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Step 11.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot - 1}}{2 \cdot 1}$

Step 11.2.5.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{9 \pm \sqrt{81 + 4}}{2 \cdot 1}$

Step 11.2.5.1.3

Add $81$ and $4$ .

$x = \frac{9 \pm \sqrt{85}}{2 \cdot 1}$

Step 11.2.5.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{85}}{2}$

Step 11.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{9 - \sqrt{85}}{2}$

Step 11.2.6

The final answer is the combination of both solutions.

$x = \frac{9 + \sqrt{85}}{2}, \frac{9 - \sqrt{85}}{2}$

Step 12

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

$x = - \frac{1}{2}, \frac{9 + \sqrt{85}}{2}, \frac{9 - \sqrt{85}}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}, \frac{9 + \sqrt{85}}{2}, \frac{9 - \sqrt{85}}{2}$

Decimal Form:

$x = - 0.5, 9.10977222 \dots, - 0.10977222 \dots$

Step 14