Linear Algebra Examples

$\frac{30 x^{3} - 63 x^{2} + 45 x}{4 x^{4} - 8 x^{3} + 7 x^{2} - 12 x + 9}$

Step 1

Set the denominator in $\frac{30 x^{3} - 63 x^{2} + 45 x}{4 x^{4} - 8 x^{3} + 7 x^{2} - 12 x + 9}$ equal to $0$ to find where the expression is undefined.

$4 x^{4} - 8 x^{3} + 7 x^{2} - 12 x + 9 = 0$

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 $4 x^{4} - 8 x^{3} + 7 x^{2} - 12 x + 9$ 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.

$p = \pm 1, \pm 9, \pm 3$

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

Step 2.1.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 9, \pm 2.25, \pm 4.5, \pm 3, \pm 0.75, \pm 1.5$

Step 2.1.1.3

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 2.1.1.3.1

Substitute $1$ into the polynomial.

$4 \cdot 1^{4} - 8 \cdot 1^{3} + 7 \cdot 1^{2} - 12 \cdot 1 + 9$

Step 2.1.1.3.2

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

$4 \cdot 1 - 8 \cdot 1^{3} + 7 \cdot 1^{2} - 12 \cdot 1 + 9$

Step 2.1.1.3.3

Multiply $4$ by $1$ .

$4 - 8 \cdot 1^{3} + 7 \cdot 1^{2} - 12 \cdot 1 + 9$

Step 2.1.1.3.4

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

$4 - 8 \cdot 1 + 7 \cdot 1^{2} - 12 \cdot 1 + 9$

Step 2.1.1.3.5

Multiply $- 8$ by $1$ .

$4 - 8 + 7 \cdot 1^{2} - 12 \cdot 1 + 9$

Step 2.1.1.3.6

Subtract $8$ from $4$ .

$- 4 + 7 \cdot 1^{2} - 12 \cdot 1 + 9$

Step 2.1.1.3.7

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

$- 4 + 7 \cdot 1 - 12 \cdot 1 + 9$

Step 2.1.1.3.8

Multiply $7$ by $1$ .

$- 4 + 7 - 12 \cdot 1 + 9$

Step 2.1.1.3.9

Add $- 4$ and $7$ .

$3 - 12 \cdot 1 + 9$

Step 2.1.1.3.10

Multiply $- 12$ by $1$ .

$3 - 12 + 9$

Step 2.1.1.3.11

Subtract $12$ from $3$ .

$- 9 + 9$

Step 2.1.1.3.12

Add $- 9$ and $9$ .

$0$

Step 2.1.1.4

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.

$\frac{4 x^{4} - 8 x^{3} + 7 x^{2} - 12 x + 9}{x - 1}$

Step 2.1.1.5

Divide $4 x^{4} - 8 x^{3} + 7 x^{2} - 12 x + 9$ by $x - 1$ .

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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 $0$ .

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

Step 2.1.1.5.2

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

$4 x^{3}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

$4 x^{3}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

Step 2.1.1.5.4

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

$4 x^{3}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

Step 2.1.1.5.5

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

$4 x^{3}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

Step 2.1.1.5.6

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

$4 x^{3}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

Step 2.1.1.5.7

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

$4 x^{3}$

$4 x^{2}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

$4 x^{3}$

$4 x^{2}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 2.1.1.5.9

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

$4 x^{3}$

$4 x^{2}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 2.1.1.5.10

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

$4 x^{3}$

$4 x^{2}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

Step 2.1.1.5.11

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

$4 x^{3}$

$4 x^{2}$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

Step 2.1.1.5.12

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

$4 x^{3}$

$4 x^{2}$

$3 x$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

$4 x^{3}$

$4 x^{2}$

$3 x$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

$3 x^{2}$

$3 x$

Step 2.1.1.5.14

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

$4 x^{3}$

$4 x^{2}$

$3 x$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

$3 x^{2}$

$3 x$

Step 2.1.1.5.15

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

$4 x^{3}$

$4 x^{2}$

$3 x$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

$3 x^{2}$

$3 x$

$9 x$

Step 2.1.1.5.16

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

$4 x^{3}$

$4 x^{2}$

$3 x$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

$3 x^{2}$

$3 x$

$9 x$

$9$

Step 2.1.1.5.17

Divide the highest order term in the dividend $- 9 x$ by the highest order term in divisor $x$ .

$4 x^{3}$

$4 x^{2}$

$3 x$

$9$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

$3 x^{2}$

$3 x$

$9 x$

$9$

Step 2.1.1.5.18

Multiply the new quotient term by the divisor.

$4 x^{3}$

$4 x^{2}$

$3 x$

$9$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

$3 x^{2}$

$3 x$

$9 x$

$9$

$9 x$

$9$

Step 2.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x + 9$

$4 x^{3}$

$4 x^{2}$

$3 x$

$9$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

$3 x^{2}$

$3 x$

$9 x$

$9$

$9 x$

$9$

Step 2.1.1.5.20

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

$4 x^{3}$

$4 x^{2}$

$3 x$

$9$

$x$

$1$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$12 x$

$9$

$4 x^{4}$

$4 x^{3}$

$7 x^{2}$

$4 x^{3}$

$4 x^{2}$

$3 x^{2}$

$12 x$

$3 x^{2}$

$3 x$

$9 x$

$9$

$9 x$

$9$

$0$

Step 2.1.1.5.21

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

$4 x^{3} - 4 x^{2} + 3 x - 9$

Step 2.1.1.6

Write $4 x^{4} - 8 x^{3} + 7 x^{2} - 12 x + 9$ as a set of factors.

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

Step 2.1.2

Factor $4 x^{3} - 4 x^{2} + 3 x - 9$ using the rational roots test.

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Step 2.1.2.1

Factor $4 x^{3} - 4 x^{2} + 3 x - 9$ 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.

$p = \pm 1, \pm 9, \pm 3$

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

Step 2.1.2.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 9, \pm 2.25, \pm 4.5, \pm 3, \pm 0.75, \pm 1.5$

Step 2.1.2.1.3

Substitute $1.5$ and simplify the expression. In this case, the expression is equal to $0$ so $1.5$ is a root of the polynomial.

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Step 2.1.2.1.3.1

Substitute $1.5$ into the polynomial.

$4 \cdot {1.5}^{3} - 4 \cdot {1.5}^{2} + 3 \cdot 1.5 - 9$

Step 2.1.2.1.3.2

Raise $1.5$ to the power of $3$ .

$4 \cdot 3.375 - 4 \cdot {1.5}^{2} + 3 \cdot 1.5 - 9$

Step 2.1.2.1.3.3

Multiply $4$ by $3.375$ .

$13.5 - 4 \cdot {1.5}^{2} + 3 \cdot 1.5 - 9$

Step 2.1.2.1.3.4

Raise $1.5$ to the power of $2$ .

$13.5 - 4 \cdot 2.25 + 3 \cdot 1.5 - 9$

Step 2.1.2.1.3.5

Multiply $- 4$ by $2.25$ .

$13.5 - 9 + 3 \cdot 1.5 - 9$

Step 2.1.2.1.3.6

Subtract $9$ from $13.5$ .

$4.5 + 3 \cdot 1.5 - 9$

Step 2.1.2.1.3.7

Multiply $3$ by $1.5$ .

$4.5 + 4.5 - 9$

Step 2.1.2.1.3.8

Add $4.5$ and $4.5$ .

$9 - 9$

Step 2.1.2.1.3.9

Subtract $9$ from $9$ .

$0$

Step 2.1.2.1.4

Since $1.5$ is a known root, divide the polynomial by $2 x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 4 x^{2} + 3 x - 9}{2 x - 3}$

Step 2.1.2.1.5

Divide $4 x^{3} - 4 x^{2} + 3 x - 9$ by $2 x - 3$ .

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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 $0$ .

$2 x$

$3$

$4 x^{3}$

$4 x^{2}$

$3 x$

$9$

Step 2.1.2.1.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$

Step 2.1.2.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			+	$4 x^{3}$	-	$6 x^{2}$

Step 2.1.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 6 x^{2}$

				$2 x^{2}$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$

Step 2.1.2.1.5.5

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

				$2 x^{2}$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$

Step 2.1.2.1.5.6

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

				$2 x^{2}$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$

Step 2.1.2.1.5.7

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

				$2 x^{2}$	+	$x$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$

Step 2.1.2.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$3 x$

Step 2.1.2.1.5.9

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

				$2 x^{2}$	+	$x$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$3 x$

Step 2.1.2.1.5.10

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

				$2 x^{2}$	+	$x$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$3 x$
							+	$6 x$

Step 2.1.2.1.5.11

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

				$2 x^{2}$	+	$x$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$3 x$
							+	$6 x$	-	$9$

Step 2.1.2.1.5.12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	+	$x$	+	$3$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$3 x$
							+	$6 x$	-	$9$

Step 2.1.2.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$	+	$3$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$3 x$
							+	$6 x$	-	$9$
							+	$6 x$	-	$9$

Step 2.1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x - 9$

				$2 x^{2}$	+	$x$	+	$3$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$3 x$
							+	$6 x$	-	$9$
							-	$6 x$	+	$9$

Step 2.1.2.1.5.15

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

				$2 x^{2}$	+	$x$	+	$3$
$2 x$	-	$3$		$4 x^{3}$	-	$4 x^{2}$	+	$3 x$	-	$9$
			-	$4 x^{3}$	+	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$3 x$
							+	$6 x$	-	$9$
							-	$6 x$	+	$9$
										$0$

Step 2.1.2.1.5.16

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

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

Step 2.1.2.1.6

Write $4 x^{3} - 4 x^{2} + 3 x - 9$ as a set of factors.

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

Step 2.1.2.2

Remove unnecessary parentheses.

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

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 $0$ .

$x - 1 = 0$

$2 x - 3 = 0$

$2 x^{2} + x + 3 = 0$

Step 2.3

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

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Step 2.3.1

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

$x - 1 = 0$

Step 2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.4

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 2.4.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 2.4.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 2.4.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 2.4.2.2.1

Divide each term in $2 x = 3$ by $2$ .

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

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.5

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

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Step 2.5.1

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

$2 x^{2} + x + 3 = 0$

Step 2.5.2

Solve $2 x^{2} + x + 3 = 0$ 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 $a = 2$ , $b = 1$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (2 \cdot 3)}}{2 \cdot 2}$

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

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot 3}}{2 \cdot 2}$

Step 2.5.2.3.1.2.2

Multiply $- 8$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 24}}{2 \cdot 2}$

Step 2.5.2.3.1.3

Subtract $24$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 2.5.2.3.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 2.5.2.3.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 2.5.2.3.1.6

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

$x = \frac{- 1 \pm i \sqrt{23}}{2 \cdot 2}$

Step 2.5.2.3.2

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm i \sqrt{23}}{4}$

Step 2.5.2.4

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

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Step 2.5.2.4.1

Simplify the numerator.

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Step 2.5.2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 2.5.2.4.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 2.5.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot 3}}{2 \cdot 2}$

Step 2.5.2.4.1.2.2

Multiply $- 8$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 24}}{2 \cdot 2}$

Step 2.5.2.4.1.3

Subtract $24$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 2.5.2.4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 2.5.2.4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 2.5.2.4.1.6

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

$x = \frac{- 1 \pm i \sqrt{23}}{2 \cdot 2}$

Step 2.5.2.4.2

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm i \sqrt{23}}{4}$

Step 2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{23}}{4}$

Step 2.5.2.4.4

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

$x = \frac{- 1 \cdot 1 + i \sqrt{23}}{4}$

Step 2.5.2.4.5

Factor $- 1$ out of $i \sqrt{23}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{23})}{4}$

Step 2.5.2.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{23})$ .

$x = \frac{- 1 (1 - i \sqrt{23})}{4}$

Step 2.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{23}}{4}$

Step 2.5.2.5

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

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Step 2.5.2.5.1

Simplify the numerator.

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Step 2.5.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 2.5.2.5.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 2.5.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot 3}}{2 \cdot 2}$

Step 2.5.2.5.1.2.2

Multiply $- 8$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 24}}{2 \cdot 2}$

Step 2.5.2.5.1.3

Subtract $24$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 2.5.2.5.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 2.5.2.5.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 2.5.2.5.1.6

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

$x = \frac{- 1 \pm i \sqrt{23}}{2 \cdot 2}$

Step 2.5.2.5.2

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm i \sqrt{23}}{4}$

Step 2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{23}}{4}$

Step 2.5.2.5.4

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

$x = \frac{- 1 \cdot 1 - i \sqrt{23}}{4}$

Step 2.5.2.5.5

Factor $- 1$ out of $- i \sqrt{23}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{23})}{4}$

Step 2.5.2.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{23})$ .

$x = \frac{- 1 (1 + i \sqrt{23})}{4}$

Step 2.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{23}}{4}$

Step 2.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{23}}{4}, - \frac{1 + i \sqrt{23}}{4}$

Step 2.6

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

$x = 1, \frac{3}{2}, - \frac{1 - i \sqrt{23}}{4}, - \frac{1 + i \sqrt{23}}{4}$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 1) \cup (1, \frac{3}{2}) \cup (\frac{3}{2}, \infty)$

Set-Builder Notation:

${x | x \neq 1, \frac{3}{2}}$

Step 4