Precalculus Examples

$f (x) = 3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116$

Step 1

Set $3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116$ equal to $0$ .

$3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116$ using the rational roots test.

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Step 2.1.1.1

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.

$p = \pm 1, \pm 116, \pm 2, \pm 58, \pm 4, \pm 29$

$q = \pm 1, \pm 3$

Step 2.1.1.2

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

$\pm 1, \pm 0.\overline{3}, \pm 116, \pm 38.\overline{6}, \pm 2, \pm 0.\overline{6}, \pm 58, \pm 19.\overline{3}, \pm 4, \pm 1.\overline{3}, \pm 29, \pm 9.\overline{6}$

Step 2.1.1.3

Substitute $0.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{3}$ is a root of the polynomial.

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Step 2.1.1.3.1

Substitute $0.\overline{3}$ into the polynomial.

$3 \cdot {0.\overline{3}}^{4} - 19 \cdot {0.\overline{3}}^{3} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Step 2.1.1.3.2

Raise $0.\overline{3}$ to the power of $4$ .

$3 \cdot 0.01234567 - 19 \cdot {0.\overline{3}}^{3} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Step 2.1.1.3.3

Multiply $3$ by $0.01234567$ .

$0.\overline{037} - 19 \cdot {0.\overline{3}}^{3} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Step 2.1.1.3.4

Raise $0.\overline{3}$ to the power of $3$ .

$0.\overline{037} - 19 \cdot 0.\overline{037} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Step 2.1.1.3.5

Multiply $- 19$ by $0.\overline{037}$ .

$0.\overline{037} - 0.\overline{703} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Step 2.1.1.3.6

Subtract $0.\overline{703}$ from $0.\overline{037}$ .

$- 0.66666666 - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Step 2.1.1.3.7

Raise $0.\overline{3}$ to the power of $2$ .

$- 0.66666666 - 27 \cdot 0.\overline{1} + 359 \cdot 0.\overline{3} - 116$

Step 2.1.1.3.8

Multiply $- 27$ by $0.\overline{1}$ .

$- 0.66666666 - 3 + 359 \cdot 0.\overline{3} - 116$

Step 2.1.1.3.9

Subtract $3$ from $- 0.66666666$ .

$- 3.66666666 + 359 \cdot 0.\overline{3} - 116$

Step 2.1.1.3.10

Multiply $359$ by $0.\overline{3}$ .

$- 3.66666666 + 119.\overline{6} - 116$

Step 2.1.1.3.11

Add $- 3.66666666$ and $119.\overline{6}$ .

$116 - 116$

Step 2.1.1.3.12

Subtract $116$ from $116$ .

$0$

Step 2.1.1.4

Since $0.\overline{3}$ is a known root, divide the polynomial by $3 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116}{3 x - 1}$

Step 2.1.1.5

Divide $3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116$ by $3 x - 1$ .

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Step 2.1.1.5.1

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

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

Step 2.1.1.5.2

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

					$x^{3}$
	$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{3}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			+	$3 x^{4}$	-	$x^{3}$

Step 2.1.1.5.4

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

				$x^{3}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$

Step 2.1.1.5.5

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

				$x^{3}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$

Step 2.1.1.5.6

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

				$x^{3}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$

Step 2.1.1.5.7

Divide the highest order term in the dividend $- 18 x^{3}$ by the highest order term in divisor $3 x$ .

				$x^{3}$	-	$6 x^{2}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$6 x^{2}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					-	$18 x^{3}$	+	$6 x^{2}$

Step 2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{3} + 6 x^{2}$

				$x^{3}$	-	$6 x^{2}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$

Step 2.1.1.5.10

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

				$x^{3}$	-	$6 x^{2}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$
							-	$33 x^{2}$

Step 2.1.1.5.11

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

$x^{3}$

$6 x^{2}$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

Step 2.1.1.5.12

Divide the highest order term in the dividend $- 33 x^{2}$ by the highest order term in divisor $3 x$ .

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 33 x^{2} + 11 x$

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

Step 2.1.1.5.15

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

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

$348 x$

Step 2.1.1.5.16

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

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

$348 x$

$116$

Step 2.1.1.5.17

Divide the highest order term in the dividend $348 x$ by the highest order term in divisor $3 x$ .

$x^{3}$

$6 x^{2}$

$11 x$

$116$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

$348 x$

$116$

Step 2.1.1.5.18

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$
							-	$33 x^{2}$	+	$359 x$
							+	$33 x^{2}$	-	$11 x$
									+	$348 x$	-	$116$
									+	$348 x$	-	$116$

Step 2.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $348 x - 116$

				$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$
							-	$33 x^{2}$	+	$359 x$
							+	$33 x^{2}$	-	$11 x$
									+	$348 x$	-	$116$
									-	$348 x$	+	$116$

Step 2.1.1.5.20

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

				$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$
							-	$33 x^{2}$	+	$359 x$
							+	$33 x^{2}$	-	$11 x$
									+	$348 x$	-	$116$
									-	$348 x$	+	$116$
												$0$

Step 2.1.1.5.21

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

$x^{3} - 6 x^{2} - 11 x + 116$

Step 2.1.1.6

Write $3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116$ as a set of factors.

$(3 x - 1) (x^{3} - 6 x^{2} - 11 x + 116) = 0$

Step 2.1.2

Factor $x^{3} - 6 x^{2} - 11 x + 116$ using the rational roots test.

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Step 2.1.2.1

Factor $x^{3} - 6 x^{2} - 11 x + 116$ using the rational roots test.

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Step 2.1.2.1.1

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.

$p = \pm 1, \pm 116, \pm 2, \pm 58, \pm 4, \pm 29$

$q = \pm 1$

Step 2.1.2.1.2

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

$\pm 1, \pm 116, \pm 2, \pm 58, \pm 4, \pm 29$

Step 2.1.2.1.3

Substitute $- 4$ and simplify the expression. In this case, the expression is equal to $0$ so $- 4$ is a root of the polynomial.

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Step 2.1.2.1.3.1

Substitute $- 4$ into the polynomial.

${(- 4)}^{3} - 6 {(- 4)}^{2} - 11 \cdot - 4 + 116$

Step 2.1.2.1.3.2

Raise $- 4$ to the power of $3$ .

$- 64 - 6 {(- 4)}^{2} - 11 \cdot - 4 + 116$

Step 2.1.2.1.3.3

Raise $- 4$ to the power of $2$ .

$- 64 - 6 \cdot 16 - 11 \cdot - 4 + 116$

Step 2.1.2.1.3.4

Multiply $- 6$ by $16$ .

$- 64 - 96 - 11 \cdot - 4 + 116$

Step 2.1.2.1.3.5

Subtract $96$ from $- 64$ .

$- 160 - 11 \cdot - 4 + 116$

Step 2.1.2.1.3.6

Multiply $- 11$ by $- 4$ .

$- 160 + 44 + 116$

Step 2.1.2.1.3.7

Add $- 160$ and $44$ .

$- 116 + 116$

Step 2.1.2.1.3.8

Add $- 116$ and $116$ .

$0$

Step 2.1.2.1.4

Since $- 4$ is a known root, divide the polynomial by $x + 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 6 x^{2} - 11 x + 116}{x + 4}$

Step 2.1.2.1.5

Divide $x^{3} - 6 x^{2} - 11 x + 116$ by $x + 4$ .

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Step 2.1.2.1.5.1

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

$x$

$4$

$x^{3}$

$6 x^{2}$

$11 x$

$116$

Step 2.1.2.1.5.2

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

					$x^{2}$
	$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$

Step 2.1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			+	$x^{3}$	+	$4 x^{2}$

Step 2.1.2.1.5.4

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$

Step 2.1.2.1.5.5

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$

Step 2.1.2.1.5.6

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$

Step 2.1.2.1.5.7

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$

Step 2.1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					-	$10 x^{2}$	-	$40 x$

Step 2.1.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 40 x$

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$

Step 2.1.2.1.5.10

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

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$

Step 2.1.2.1.5.11

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

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$

Step 2.1.2.1.5.12

Divide the highest order term in the dividend $29 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$	+	$29$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$

Step 2.1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$	+	$29$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$
							+	$29 x$	+	$116$

Step 2.1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $29 x + 116$

				$x^{2}$	-	$10 x$	+	$29$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$
							-	$29 x$	-	$116$

Step 2.1.2.1.5.15

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

				$x^{2}$	-	$10 x$	+	$29$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$
							-	$29 x$	-	$116$
										$0$

Step 2.1.2.1.5.16

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

$x^{2} - 10 x + 29$

Step 2.1.2.1.6

Write $x^{3} - 6 x^{2} - 11 x + 116$ as a set of factors.

$(3 x - 1) ((x + 4) (x^{2} - 10 x + 29)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$(3 x - 1) (x + 4) (x^{2} - 10 x + 29) = 0$

Step 2.2

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

$3 x - 1 = 0$

$x + 4 = 0$

$x^{2} - 10 x + 29 = 0$

Step 2.3

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 2.3.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 2.3.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 2.3.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 2.3.2.2.1

Divide each term in $3 x = 1$ by $3$ .

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

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

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

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.4

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.5

Set $x^{2} - 10 x + 29$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} - 10 x + 29$ equal to $0$ .

$x^{2} - 10 x + 29 = 0$

Step 2.5.2

Solve $x^{2} - 10 x + 29 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values $a = 1$ , $b = - 10$ , and $c = 29$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 29)}}{2 \cdot 1}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 29$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 29}}{2 \cdot 1}$

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $29$ .

$x = \frac{10 \pm \sqrt{100 - 116}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Subtract $116$ from $100$ .

$x = \frac{10 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 2.5.2.3.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 2.5.2.3.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 2.5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{10 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 2.5.2.3.1.7

Rewrite $16$ as $4^{2}$ .

$x = \frac{10 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 2.5.2.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{10 \pm i \cdot 4}{2 \cdot 1}$

Step 2.5.2.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{10 \pm 4 i}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{10 \pm 4 i}{2}$

Step 2.5.2.3.3

Simplify $\frac{10 \pm 4 i}{2}$ .

$x = 5 \pm 2 i$

Step 2.5.2.4

The final answer is the combination of both solutions.

$x = 5 + 2 i, 5 - 2 i$

Step 2.6

The final solution is all the values that make $(3 x - 1) (x + 4) (x^{2} - 10 x + 29) = 0$ true.

$x = \frac{1}{3}, - 4, 5 + 2 i, 5 - 2 i$

Step 3