Precalculus Examples

$4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$

Step 1

Set $4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$ equal to $0$ .

$4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9 = 0$

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 $4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$ 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 $- 1.5$ 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 2.1.1.3.1

Substitute $- 1.5$ into the polynomial.

$4 {(- 1.5)}^{4} + 8 {(- 1.5)}^{3} - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 2.1.1.3.2

Raise $- 1.5$ to the power of $4$ .

$4 \cdot 5.0625 + 8 {(- 1.5)}^{3} - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 2.1.1.3.3

Multiply $4$ by $5.0625$ .

$20.25 + 8 {(- 1.5)}^{3} - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 2.1.1.3.4

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

$20.25 + 8 \cdot - 3.375 - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 2.1.1.3.5

Multiply $8$ by $- 3.375$ .

$20.25 - 27 - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 2.1.1.3.6

Subtract $27$ from $20.25$ .

$- 6.75 - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 2.1.1.3.7

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

$- 6.75 - 7 \cdot 2.25 - 21 \cdot - 1.5 - 9$

Step 2.1.1.3.8

Multiply $- 7$ by $2.25$ .

$- 6.75 - 15.75 - 21 \cdot - 1.5 - 9$

Step 2.1.1.3.9

Subtract $15.75$ from $- 6.75$ .

$- 22.5 - 21 \cdot - 1.5 - 9$

Step 2.1.1.3.10

Multiply $- 21$ by $- 1.5$ .

$- 22.5 + 31.5 - 9$

Step 2.1.1.3.11

Add $- 22.5$ and $31.5$ .

$9 - 9$

Step 2.1.1.3.12

Subtract $9$ from $9$ .

$0$

Step 2.1.1.4

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.

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{2 x + 3}$

Step 2.1.1.5

Divide $4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$ by $2 x + 3$ .

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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 $0$ .

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

Step 2.1.1.5.2

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

$2 x^{3}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

Step 2.1.1.5.4

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

				$2 x^{3}$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$

Step 2.1.1.5.5

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

$2 x^{3}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

Step 2.1.1.5.6

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

$2 x^{3}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

Step 2.1.1.5.7

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

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

Step 2.1.1.5.9

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

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

Step 2.1.1.5.10

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

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

Step 2.1.1.5.11

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

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

Step 2.1.1.5.12

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

$2 x^{3}$

$x^{2}$

$5 x$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{3}$	+	$x^{2}$	-	$5 x$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							-	$10 x^{2}$	-	$15 x$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 15 x$

$2 x^{3}$

$x^{2}$

$5 x$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

$10 x^{2}$

$15 x$

Step 2.1.1.5.15

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

$2 x^{3}$

$x^{2}$

$5 x$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

$10 x^{2}$

$15 x$

$6 x$

Step 2.1.1.5.16

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

$2 x^{3}$

$x^{2}$

$5 x$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

$10 x^{2}$

$15 x$

$6 x$

$9$

Step 2.1.1.5.17

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

				$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							+	$10 x^{2}$	+	$15 x$
									-	$6 x$	-	$9$

Step 2.1.1.5.18

Multiply the new quotient term by the divisor.

				$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							+	$10 x^{2}$	+	$15 x$
									-	$6 x$	-	$9$
									-	$6 x$	-	$9$

Step 2.1.1.5.19

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

				$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							+	$10 x^{2}$	+	$15 x$
									-	$6 x$	-	$9$
									+	$6 x$	+	$9$

Step 2.1.1.5.20

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

				$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							+	$10 x^{2}$	+	$15 x$
									-	$6 x$	-	$9$
									+	$6 x$	+	$9$
												$0$

Step 2.1.1.5.21

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

$2 x^{3} + x^{2} - 5 x - 3$

Step 2.1.1.6

Write $4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$ as a set of factors.

$(2 x + 3) (2 x^{3} + x^{2} - 5 x - 3) = 0$

Step 2.1.2

Factor $2 x^{3} + x^{2} - 5 x - 3$ using the rational roots test.

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Step 2.1.2.1

Factor $2 x^{3} + x^{2} - 5 x - 3$ 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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 2$

Step 2.1.2.1.2

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

$\pm 1, \pm 0.5, \pm 3, \pm 1.5$

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.

$2 {(- 1.5)}^{3} + {(- 1.5)}^{2} - 5 \cdot - 1.5 - 3$

Step 2.1.2.1.3.2

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

$2 \cdot - 3.375 + {(- 1.5)}^{2} - 5 \cdot - 1.5 - 3$

Step 2.1.2.1.3.3

Multiply $2$ by $- 3.375$ .

$- 6.75 + {(- 1.5)}^{2} - 5 \cdot - 1.5 - 3$

Step 2.1.2.1.3.4

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

$- 6.75 + 2.25 - 5 \cdot - 1.5 - 3$

Step 2.1.2.1.3.5

Add $- 6.75$ and $2.25$ .

$- 4.5 - 5 \cdot - 1.5 - 3$

Step 2.1.2.1.3.6

Multiply $- 5$ by $- 1.5$ .

$- 4.5 + 7.5 - 3$

Step 2.1.2.1.3.7

Add $- 4.5$ and $7.5$ .

$3 - 3$

Step 2.1.2.1.3.8

Subtract $3$ from $3$ .

$0$

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.

$\frac{2 x^{3} + x^{2} - 5 x - 3}{2 x + 3}$

Step 2.1.2.1.5

Divide $2 x^{3} + x^{2} - 5 x - 3$ 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 $0$ .

$2 x$

$3$

$2 x^{3}$

$x^{2}$

$5 x$

$3$

Step 2.1.2.1.5.2

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

					$x^{2}$
	$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$

Step 2.1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			+	$2 x^{3}$	+	$3 x^{2}$

Step 2.1.2.1.5.4

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

				$x^{2}$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$

Step 2.1.2.1.5.5

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

				$x^{2}$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$

Step 2.1.2.1.5.6

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

				$x^{2}$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$

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 $2 x$ .

				$x^{2}$	-	$x$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$

Step 2.1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					-	$2 x^{2}$	-	$3 x$

Step 2.1.2.1.5.9

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

				$x^{2}$	-	$x$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$

Step 2.1.2.1.5.10

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

				$x^{2}$	-	$x$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$

Step 2.1.2.1.5.11

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

				$x^{2}$	-	$x$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$	-	$3$

Step 2.1.2.1.5.12

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

				$x^{2}$	-	$x$	-	$1$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$	-	$3$

Step 2.1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$1$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$	-	$3$
							-	$2 x$	-	$3$

Step 2.1.2.1.5.14

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

				$x^{2}$	-	$x$	-	$1$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$	-	$3$
							+	$2 x$	+	$3$

Step 2.1.2.1.5.15

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

				$x^{2}$	-	$x$	-	$1$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$	-	$3$
							+	$2 x$	+	$3$
										$0$

Step 2.1.2.1.5.16

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

$x^{2} - x - 1$

Step 2.1.2.1.6

Write $2 x^{3} + x^{2} - 5 x - 3$ as a set of factors.

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

Step 2.1.2.2

Remove unnecessary parentheses.

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

Step 2.1.3

Combine like factors.

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Step 2.1.3.1

Raise $2 x + 3$ to the power of $1$ .

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

Step 2.1.3.2

Raise $2 x + 3$ to the power of $1$ .

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

Step 2.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.3.4

Add $1$ and $1$ .

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

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 $0$ .

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Solve ${(2 x + 3)}^{2} = 0$ for $x$ .

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Step 2.3.2.1

Set the $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 2.3.2.2

Solve for $x$ .

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Step 2.3.2.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 2.3.2.2.2

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

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Step 2.3.2.2.2.1

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

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

Step 2.3.2.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.2.2.1.1

Cancel the common factor.

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

Step 2.3.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.2.2.2.3

Simplify the right side.

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Step 2.3.2.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

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

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

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

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

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

Step 2.4.2.3.1.2

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

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 2.4.2.3.1.3

Add $1$ and $4$ .

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

Step 2.4.2.3.2

Multiply $2$ by $1$ .

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

Step 2.4.2.4

The final answer is the combination of both solutions.

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

Step 2.5

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.5, 1.61803398 \dots, - 0.61803398 \dots$

Mixed Number Form:

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

Step 4