Calculus Examples

$y = x^{4} - 8 x^{3} + 24 x^{2}$ , $y = 19 x + 52$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$x^{4} - 8 x^{3} + 24 x^{2} = 19 x + 52$

Step 1.2

Solve $x^{4} - 8 x^{3} + 24 x^{2} = 19 x + 52$ for $x$ .

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Step 1.2.1

Subtract $19 x$ from both sides of the equation.

$x^{4} - 8 x^{3} + 24 x^{2} - 19 x = 52$

Step 1.2.2

Subtract $52$ from both sides of the equation.

$x^{4} - 8 x^{3} + 24 x^{2} - 19 x - 52 = 0$

Step 1.2.3

Factor the left side of the equation.

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Step 1.2.3.1

Factor $x^{4} - 8 x^{3} + 24 x^{2} - 19 x - 52$ using the rational roots test.

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Step 1.2.3.1.1

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.

$p = \pm 1, \pm 52, \pm 2, \pm 26, \pm 4, \pm 13$

$q = \pm 1 + y = 19 x + 52$

Step 1.2.3.1.2

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

$\pm 1, \pm 52, \pm 2, \pm 26, \pm 4, \pm 13 + y = 19 x + 52$

Step 1.2.3.1.3

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 1.2.3.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - 8 {(- 1)}^{3} + 24 {(- 1)}^{2} - 19 \cdot - 1 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.2

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

$1 - 8 {(- 1)}^{3} + 24 {(- 1)}^{2} - 19 \cdot - 1 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.3

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

$1 - 8 \cdot - 1 + 24 {(- 1)}^{2} - 19 \cdot - 1 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.4

Multiply $- 8$ by $- 1$ .

$1 + 8 + 24 {(- 1)}^{2} - 19 \cdot - 1 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.5

Add $1$ and $8$ .

$9 + 24 {(- 1)}^{2} - 19 \cdot - 1 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.6

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

$9 + 24 \cdot 1 - 19 \cdot - 1 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.7

Multiply $24$ by $1$ .

$9 + 24 - 19 \cdot - 1 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.8

Add $9$ and $24$ .

$33 - 19 \cdot - 1 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.9

Multiply $- 19$ by $- 1$ .

$33 + 19 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.10

Add $33$ and $19$ .

$52 - 52 + y = 19 x + 52$

Step 1.2.3.1.3.11

Subtract $52$ from $52$ .

$0 + y = 19 x + 52$

Step 1.2.3.1.4

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.

$\frac{x^{4} - 8 x^{3} + 24 x^{2} - 19 x - 52}{x + 1} + y = 19 x + 52$

Step 1.2.3.1.5

Divide $x^{4} - 8 x^{3} + 24 x^{2} - 19 x - 52$ by $x + 1$ .

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Step 1.2.3.1.5.1

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

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

Step 1.2.3.1.5.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

Step 1.2.3.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

Step 1.2.3.1.5.4

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$8 x^{3}$	+	$24 x^{2}$	-	$19 x$	-	$52$
			-	$x^{4}$	-	$x^{3}$

Step 1.2.3.1.5.5

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$8 x^{3}$	+	$24 x^{2}$	-	$19 x$	-	$52$
			-	$x^{4}$	-	$x^{3}$
					-	$9 x^{3}$

Step 1.2.3.1.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

Step 1.2.3.1.5.7

Divide the highest order term in the dividend $- 9 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$9 x^{2}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

Step 1.2.3.1.5.8

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$9 x^{2}$
$x$	+	$1$		$x^{4}$	-	$8 x^{3}$	+	$24 x^{2}$	-	$19 x$	-	$52$
			-	$x^{4}$	-	$x^{3}$
					-	$9 x^{3}$	+	$24 x^{2}$
					-	$9 x^{3}$	-	$9 x^{2}$

Step 1.2.3.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{3} - 9 x^{2}$

$x^{3}$

$9 x^{2}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

Step 1.2.3.1.5.10

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

$x^{3}$

$9 x^{2}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

Step 1.2.3.1.5.11

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

$x^{3}$

$9 x^{2}$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

$19 x$

Step 1.2.3.1.5.12

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

$x^{3}$

$9 x^{2}$

$33 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

$19 x$

Step 1.2.3.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$9 x^{2}$

$33 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

$19 x$

$33 x^{2}$

$33 x$

Step 1.2.3.1.5.14

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

$x^{3}$

$9 x^{2}$

$33 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

$19 x$

$33 x^{2}$

$33 x$

Step 1.2.3.1.5.15

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

$x^{3}$

$9 x^{2}$

$33 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

$19 x$

$33 x^{2}$

$33 x$

$52 x$

Step 1.2.3.1.5.16

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

$x^{3}$

$9 x^{2}$

$33 x$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

$19 x$

$33 x^{2}$

$33 x$

$52 x$

$52$

Step 1.2.3.1.5.17

Divide the highest order term in the dividend $- 52 x$ by the highest order term in divisor $x$ .

$x^{3}$

$9 x^{2}$

$33 x$

$52$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

$19 x$

$33 x^{2}$

$33 x$

$52 x$

$52$

Step 1.2.3.1.5.18

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
$x$	+	$1$		$x^{4}$	-	$8 x^{3}$	+	$24 x^{2}$	-	$19 x$	-	$52$
			-	$x^{4}$	-	$x^{3}$
					-	$9 x^{3}$	+	$24 x^{2}$
					+	$9 x^{3}$	+	$9 x^{2}$
							+	$33 x^{2}$	-	$19 x$
							-	$33 x^{2}$	-	$33 x$
									-	$52 x$	-	$52$
									-	$52 x$	-	$52$

Step 1.2.3.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 52 x - 52$

$x^{3}$

$9 x^{2}$

$33 x$

$52$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

$19 x$

$33 x^{2}$

$33 x$

$52 x$

$52$

$52 x$

$52$

Step 1.2.3.1.5.20

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

$x^{3}$

$9 x^{2}$

$33 x$

$52$

$x$

$1$

$x^{4}$

$8 x^{3}$

$24 x^{2}$

$19 x$

$52$

$x^{4}$

$x^{3}$

$9 x^{3}$

$24 x^{2}$

$9 x^{3}$

$9 x^{2}$

$33 x^{2}$

$19 x$

$33 x^{2}$

$33 x$

$52 x$

$52$

$52 x$

$52$

$0$

Step 1.2.3.1.5.21

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

$x^{3} - 9 x^{2} + 33 x - 52 + y = 19 x + 52$

Step 1.2.3.1.6

Write $x^{4} - 8 x^{3} + 24 x^{2} - 19 x - 52$ as a set of factors.

$(x + 1) (x^{3} - 9 x^{2} + 33 x - 52) = 0$

Step 1.2.3.2

Factor $x^{3} - 9 x^{2} + 33 x - 52$ using the rational roots test.

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Step 1.2.3.2.1

Factor $x^{3} - 9 x^{2} + 33 x - 52$ using the rational roots test.

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Step 1.2.3.2.1.1

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.

$p = \pm 1, \pm 52, \pm 2, \pm 26, \pm 4, \pm 13$

$q = \pm 1 + y = 19 x + 52$

Step 1.2.3.2.1.2

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

$\pm 1, \pm 52, \pm 2, \pm 26, \pm 4, \pm 13 + y = 19 x + 52$

Step 1.2.3.2.1.3

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 1.2.3.2.1.3.1

Substitute $4$ into the polynomial.

$4^{3} - 9 \cdot 4^{2} + 33 \cdot 4 - 52 + y = 19 x + 52$

Step 1.2.3.2.1.3.2

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

$64 - 9 \cdot 4^{2} + 33 \cdot 4 - 52 + y = 19 x + 52$

Step 1.2.3.2.1.3.3

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

$64 - 9 \cdot 16 + 33 \cdot 4 - 52 + y = 19 x + 52$

Step 1.2.3.2.1.3.4

Multiply $- 9$ by $16$ .

$64 - 144 + 33 \cdot 4 - 52 + y = 19 x + 52$

Step 1.2.3.2.1.3.5

Subtract $144$ from $64$ .

$- 80 + 33 \cdot 4 - 52 + y = 19 x + 52$

Step 1.2.3.2.1.3.6

Multiply $33$ by $4$ .

$- 80 + 132 - 52 + y = 19 x + 52$

Step 1.2.3.2.1.3.7

Add $- 80$ and $132$ .

$52 - 52 + y = 19 x + 52$

Step 1.2.3.2.1.3.8

Subtract $52$ from $52$ .

$0 + y = 19 x + 52$

Step 1.2.3.2.1.4

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.

$\frac{x^{3} - 9 x^{2} + 33 x - 52}{x - 4} + y = 19 x + 52$

Step 1.2.3.2.1.5

Divide $x^{3} - 9 x^{2} + 33 x - 52$ by $x - 4$ .

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Step 1.2.3.2.1.5.1

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

$x$

$4$

$x^{3}$

$9 x^{2}$

$33 x$

$52$

Step 1.2.3.2.1.5.2

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

					$x^{2}$
	$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$

Step 1.2.3.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			+	$x^{3}$	-	$4 x^{2}$

Step 1.2.3.2.1.5.4

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$

Step 1.2.3.2.1.5.5

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$

Step 1.2.3.2.1.5.6

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$

Step 1.2.3.2.1.5.7

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$

Step 1.2.3.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$
					-	$5 x^{2}$	+	$20 x$

Step 1.2.3.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} + 20 x$

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$
					+	$5 x^{2}$	-	$20 x$

Step 1.2.3.2.1.5.10

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

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$
					+	$5 x^{2}$	-	$20 x$
							+	$13 x$

Step 1.2.3.2.1.5.11

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

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$
					+	$5 x^{2}$	-	$20 x$
							+	$13 x$	-	$52$

Step 1.2.3.2.1.5.12

Divide the highest order term in the dividend $13 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$	+	$13$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$
					+	$5 x^{2}$	-	$20 x$
							+	$13 x$	-	$52$

Step 1.2.3.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$	+	$13$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$
					+	$5 x^{2}$	-	$20 x$
							+	$13 x$	-	$52$
							+	$13 x$	-	$52$

Step 1.2.3.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $13 x - 52$

				$x^{2}$	-	$5 x$	+	$13$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$
					+	$5 x^{2}$	-	$20 x$
							+	$13 x$	-	$52$
							-	$13 x$	+	$52$

Step 1.2.3.2.1.5.15

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

				$x^{2}$	-	$5 x$	+	$13$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$33 x$	-	$52$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$33 x$
					+	$5 x^{2}$	-	$20 x$
							+	$13 x$	-	$52$
							-	$13 x$	+	$52$
										$0$

Step 1.2.3.2.1.5.16

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

$x^{2} - 5 x + 13 + y = 19 x + 52$

Step 1.2.3.2.1.6

Write $x^{3} - 9 x^{2} + 33 x - 52$ as a set of factors.

$(x + 1) ((x - 4) (x^{2} - 5 x + 13)) = 0$

Step 1.2.3.2.2

Remove unnecessary parentheses.

$(x + 1) (x - 4) (x^{2} - 5 x + 13) = 0$

Step 1.2.4

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

$x + 1 = 0$

$x - 4 = 0$

$x^{2} - 5 x + 13 = 0 + y = 19 x + 52$

Step 1.2.5

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

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Step 1.2.5.1

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

$x + 1 = 0$

Step 1.2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.6

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

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Step 1.2.6.1

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

$x - 4 = 0$

Step 1.2.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.7

Set $x^{2} - 5 x + 13$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $x^{2} - 5 x + 13$ equal to $0$ .

$x^{2} - 5 x + 13 = 0$

Step 1.2.7.2

Solve $x^{2} - 5 x + 13 = 0$ for $x$ .

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Step 1.2.7.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 19 x + 52$

Step 1.2.7.2.2

Substitute the values $a = 1$ , $b = - 5$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1} + y = 19 x + 52$

Step 1.2.7.2.3

Simplify.

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Step 1.2.7.2.3.1

Simplify the numerator.

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Step 1.2.7.2.3.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 1.2.7.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 1.2.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 13}}{2 \cdot 1}$

Step 1.2.7.2.3.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{5 \pm \sqrt{25 - 52}}{2 \cdot 1}$

Step 1.2.7.2.3.1.3

Subtract $52$ from $25$ .

$x = \frac{5 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 1.2.7.2.3.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 1.2.7.2.3.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 1.2.7.2.3.1.6

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

$x = \frac{5 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 1.2.7.2.3.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.2.7.2.3.1.7.1

Factor $9$ out of $27$ .

$x = \frac{5 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 1.2.7.2.3.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{5 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.7.2.3.1.8

Pull terms out from under the radical.

$x = \frac{5 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 1.2.7.2.3.1.9

Move $3$ to the left of $i$ .

$x = \frac{5 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 1.2.7.2.3.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm 3 i \sqrt{3}}{2}$

Step 1.2.7.2.4

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

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Step 1.2.7.2.4.1

Simplify the numerator.

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Step 1.2.7.2.4.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 1.2.7.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 1.2.7.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 13}}{2 \cdot 1}$

Step 1.2.7.2.4.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{5 \pm \sqrt{25 - 52}}{2 \cdot 1}$

Step 1.2.7.2.4.1.3

Subtract $52$ from $25$ .

$x = \frac{5 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 1.2.7.2.4.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 1.2.7.2.4.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 1.2.7.2.4.1.6

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

$x = \frac{5 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 1.2.7.2.4.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.2.7.2.4.1.7.1

Factor $9$ out of $27$ .

$x = \frac{5 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 1.2.7.2.4.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{5 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.7.2.4.1.8

Pull terms out from under the radical.

$x = \frac{5 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 1.2.7.2.4.1.9

Move $3$ to the left of $i$ .

$x = \frac{5 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 1.2.7.2.4.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm 3 i \sqrt{3}}{2}$

Step 1.2.7.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + 3 i \sqrt{3}}{2}$

Step 1.2.7.2.5

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

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Step 1.2.7.2.5.1

Simplify the numerator.

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Step 1.2.7.2.5.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 1.2.7.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 1.2.7.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 13}}{2 \cdot 1}$

Step 1.2.7.2.5.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{5 \pm \sqrt{25 - 52}}{2 \cdot 1}$

Step 1.2.7.2.5.1.3

Subtract $52$ from $25$ .

$x = \frac{5 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 1.2.7.2.5.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 1.2.7.2.5.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 1.2.7.2.5.1.6

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

$x = \frac{5 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 1.2.7.2.5.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.2.7.2.5.1.7.1

Factor $9$ out of $27$ .

$x = \frac{5 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 1.2.7.2.5.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{5 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.7.2.5.1.8

Pull terms out from under the radical.

$x = \frac{5 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 1.2.7.2.5.1.9

Move $3$ to the left of $i$ .

$x = \frac{5 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 1.2.7.2.5.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm 3 i \sqrt{3}}{2}$

Step 1.2.7.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - 3 i \sqrt{3}}{2}$

Step 1.2.7.2.6

The final answer is the combination of both solutions.

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

Step 1.2.8

The final solution is all the values that make $(x + 1) (x - 4) (x^{2} - 5 x + 13) = 0$ true.

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

Step 1.3

Evaluate $y$ when $x = - 1$ .

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Step 1.3.1

Substitute $- 1$ for $x$ .

$y = 19 (- 1) + 52$

Step 1.3.2

Simplify $19 (- 1) + 52$ .

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Step 1.3.2.1

Multiply $19$ by $- 1$ .

$y = - 19 + 52$

Step 1.3.2.2

Add $- 19$ and $52$ .

$y = 33$

Step 1.4

Evaluate $y$ when $x = 4$ .

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Step 1.4.1

Substitute $4$ for $x$ .

$y = 19 (4) + 52$

Step 1.4.2

Simplify $19 (4) + 52$ .

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Step 1.4.2.1

Multiply $19$ by $4$ .

$y = 76 + 52$

Step 1.4.2.2

Add $76$ and $52$ .

$y = 128$

Step 1.5

Evaluate $y$ when $x = \frac{5 + 3 i \sqrt{3}}{2}$ .

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Step 1.5.1

Substitute $\frac{5 + 3 i \sqrt{3}}{2}$ for $x$ .

$y = 19 (\frac{5 + 3 i \sqrt{3}}{2}) + 52$

Step 1.5.2

Substitute $\frac{5 + 3 i \sqrt{3}}{2}$ for $x$ in $y = 19 (\frac{5 + 3 i \sqrt{3}}{2}) + 52$ and solve for $y$ .

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Step 1.5.2.1

Multiply $19$ by $\frac{5 + 3 i \sqrt{3}}{2}$ .

$y = 19 (\frac{5 + 3 i \sqrt{3}}{2}) + 52$

Step 1.5.2.2

Simplify $19 (\frac{5 + 3 i \sqrt{3}}{2}) + 52$ .

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Step 1.5.2.2.1

Combine $19$ and $\frac{5 + 3 i \sqrt{3}}{2}$ .

$y = \frac{19 (5 + 3 i \sqrt{3})}{2} + 52$

Step 1.5.2.2.2

To write $52$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{19 (5 + 3 i \sqrt{3})}{2} + 52 \cdot \frac{2}{2}$

Step 1.5.2.2.3

Combine $52$ and $\frac{2}{2}$ .

$y = \frac{19 (5 + 3 i \sqrt{3})}{2} + \frac{52 \cdot 2}{2}$

Step 1.5.2.2.4

Combine the numerators over the common denominator.

$y = \frac{199 + 57 i \sqrt{3}}{2}$

Step 1.6

Evaluate $y$ when $x = \frac{5 - 3 i \sqrt{3}}{2}$ .

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Step 1.6.1

Substitute $\frac{5 - 3 i \sqrt{3}}{2}$ for $x$ .

$y = 19 (\frac{5 - 3 i \sqrt{3}}{2}) + 52$

Step 1.6.2

Substitute $\frac{5 - 3 i \sqrt{3}}{2}$ for $x$ in $y = 19 (\frac{5 - 3 i \sqrt{3}}{2}) + 52$ and solve for $y$ .

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Step 1.6.2.1

Multiply $19$ by $\frac{5 - 3 i \sqrt{3}}{2}$ .

$y = 19 (\frac{5 - 3 i \sqrt{3}}{2}) + 52$

Step 1.6.2.2

Simplify $19 (\frac{5 - 3 i \sqrt{3}}{2}) + 52$ .

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Step 1.6.2.2.1

Combine $19$ and $\frac{5 - 3 i \sqrt{3}}{2}$ .

$y = \frac{19 (5 - 3 i \sqrt{3})}{2} + 52$

Step 1.6.2.2.2

To write $52$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{19 (5 - 3 i \sqrt{3})}{2} + 52 \cdot \frac{2}{2}$

Step 1.6.2.2.3

Combine $52$ and $\frac{2}{2}$ .

$y = \frac{19 (5 - 3 i \sqrt{3})}{2} + \frac{52 \cdot 2}{2}$

Step 1.6.2.2.4

Combine the numerators over the common denominator.

$y = \frac{199 - 57 i \sqrt{3}}{2}$

Step 1.7

List all of the solutions.

$y = 33, x = - 1$

$y = 128, x = 4$

$y = \frac{199 + 57 i \sqrt{3}}{2}, x = \frac{5 + 3 i \sqrt{3}}{2}$

$y = \frac{199 - 57 i \sqrt{3}}{2}, x = \frac{5 - 3 i \sqrt{3}}{2}$

$y = 33, x = - 1$

$y = 128, x = 4$

$y = \frac{199 + 57 i \sqrt{3}}{2}, x = \frac{5 + 3 i \sqrt{3}}{2}$

$y = \frac{199 - 57 i \sqrt{3}}{2}, x = \frac{5 - 3 i \sqrt{3}}{2}$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3