Algebra Examples

$8 x^{3} - 10 x^{2} - 7 x - 1 = 0$

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.

$p = \pm 1$

$q = \pm 1, \pm 2, \pm 4, \pm 8$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}$

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.

$8 {(- \frac{1}{4})}^{3} - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - \frac{1}{4}$ 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 $- \frac{1}{4}$ .

$8 ({(- 1)}^{3} {(\frac{1}{4})}^{3}) - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4.1.1.2

Apply the product rule to $\frac{1}{4}$ .

$8 ({(- 1)}^{3} \frac{1^{3}}{4^{3}}) - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4.1.2

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

$8 (- \frac{1^{3}}{4^{3}}) - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4.1.3

One to any power is one.

$8 (- \frac{1}{4^{3}}) - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4.1.4

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

$8 (- \frac{1}{64}) - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4.1.5

Cancel the common factor of $8$ .

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Step 4.1.5.1

Move the leading negative in $- \frac{1}{64}$ into the numerator.

$8 (\frac{- 1}{64}) - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4.1.5.2

Factor $8$ out of $64$ .

$8 \frac{- 1}{8 (8)} - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4.1.5.3

Cancel the common factor.

$8 \frac{- 1}{8 \cdot 8} - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4.1.5.4

Rewrite the expression.

$\frac{- 1}{8} - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

Step 4.1.6

Move the negative in front of the fraction.

$- \frac{1}{8} - 10 {(- \frac{1}{4})}^{2} - 7 (- \frac{1}{4}) - 1$

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

$- \frac{1}{8} - 10 ({(- 1)}^{2} {(\frac{1}{4})}^{2}) - 7 (- \frac{1}{4}) - 1$

Step 4.1.7.2

Apply the product rule to $\frac{1}{4}$ .

$- \frac{1}{8} - 10 ({(- 1)}^{2} \frac{1^{2}}{4^{2}}) - 7 (- \frac{1}{4}) - 1$

Step 4.1.8

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

$- \frac{1}{8} - 10 (1 \frac{1^{2}}{4^{2}}) - 7 (- \frac{1}{4}) - 1$

Step 4.1.9

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

$- \frac{1}{8} - 10 \frac{1^{2}}{4^{2}} - 7 (- \frac{1}{4}) - 1$

Step 4.1.10

One to any power is one.

$- \frac{1}{8} - 10 \frac{1}{4^{2}} - 7 (- \frac{1}{4}) - 1$

Step 4.1.11

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

$- \frac{1}{8} - 10 (\frac{1}{16}) - 7 (- \frac{1}{4}) - 1$

Step 4.1.12

Cancel the common factor of $2$ .

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Step 4.1.12.1

Factor $2$ out of $- 10$ .

$- \frac{1}{8} + 2 (- 5) \frac{1}{16} - 7 (- \frac{1}{4}) - 1$

Step 4.1.12.2

Factor $2$ out of $16$ .

$- \frac{1}{8} + 2 \cdot - 5 \frac{1}{2 \cdot 8} - 7 (- \frac{1}{4}) - 1$

Step 4.1.12.3

Cancel the common factor.

$- \frac{1}{8} + 2 \cdot - 5 \frac{1}{2 \cdot 8} - 7 (- \frac{1}{4}) - 1$

Step 4.1.12.4

Rewrite the expression.

$- \frac{1}{8} - 5 (\frac{1}{8}) - 7 (- \frac{1}{4}) - 1$

Step 4.1.13

Combine $- 5$ and $\frac{1}{8}$ .

$- \frac{1}{8} + \frac{- 5}{8} - 7 (- \frac{1}{4}) - 1$

Step 4.1.14

Move the negative in front of the fraction.

$- \frac{1}{8} - \frac{5}{8} - 7 (- \frac{1}{4}) - 1$

Step 4.1.15

Multiply $- 7 (- \frac{1}{4})$ .

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Step 4.1.15.1

Multiply $- 1$ by $- 7$ .

$- \frac{1}{8} - \frac{5}{8} + 7 (\frac{1}{4}) - 1$

Step 4.1.15.2

Combine $7$ and $\frac{1}{4}$ .

$- \frac{1}{8} - \frac{5}{8} + \frac{7}{4} - 1$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$- 1 + \frac{- 1 - 5}{8} + \frac{7}{4}$

Step 4.2.2

Subtract $5$ from $- 1$ .

$- 1 + \frac{- 6}{8} + \frac{7}{4}$

Step 4.3

Find the common denominator.

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Step 4.3.1

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

$\frac{- 1}{1} + \frac{- 6}{8} + \frac{7}{4}$

Step 4.3.2

Multiply $\frac{- 1}{1}$ by $\frac{8}{8}$ .

$\frac{- 1}{1} \cdot \frac{8}{8} + \frac{- 6}{8} + \frac{7}{4}$

Step 4.3.3

Multiply $\frac{- 1}{1}$ by $\frac{8}{8}$ .

$\frac{- 1 \cdot 8}{8} + \frac{- 6}{8} + \frac{7}{4}$

Step 4.3.4

Multiply $\frac{7}{4}$ by $\frac{2}{2}$ .

$\frac{- 1 \cdot 8}{8} + \frac{- 6}{8} + \frac{7}{4} \cdot \frac{2}{2}$

Step 4.3.5

Multiply $\frac{7}{4}$ by $\frac{2}{2}$ .

$\frac{- 1 \cdot 8}{8} + \frac{- 6}{8} + \frac{7 \cdot 2}{4 \cdot 2}$

Step 4.3.6

Reorder the factors of $4 \cdot 2$ .

$\frac{- 1 \cdot 8}{8} + \frac{- 6}{8} + \frac{7 \cdot 2}{2 \cdot 4}$

Step 4.3.7

Multiply $2$ by $4$ .

$\frac{- 1 \cdot 8}{8} + \frac{- 6}{8} + \frac{7 \cdot 2}{8}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 1 \cdot 8 - 6 + 7 \cdot 2}{8}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $- 1$ by $8$ .

$\frac{- 8 - 6 + 7 \cdot 2}{8}$

Step 4.5.2

Multiply $7$ by $2$ .

$\frac{- 8 - 6 + 14}{8}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Subtract $6$ from $- 8$ .

$\frac{- 14 + 14}{8}$

Step 4.6.2

Add $- 14$ and $14$ .

$\frac{0}{8}$

Step 4.6.3

Divide $0$ by $8$ .

$0$

Step 5

Since $- \frac{1}{4}$ is a known root, divide the polynomial by $x + \frac{1}{4}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{8 x^{3} - 10 x^{2} - 7 x - 1}{x + \frac{1}{4}}$

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.

$- \frac{1}{4}$	$8$	$- 10$	$- 7$	$- 1$

Step 6.2

The first number in the dividend $(8)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{4}$	$8$	$- 10$	$- 7$	$- 1$

	$8$

Step 6.3

Multiply the newest entry in the result $(8)$ by the divisor $(- \frac{1}{4})$ and place the result of $(- 2)$ under the next term in the dividend $(- 10)$ .

$- \frac{1}{4}$	$8$	$- 10$	$- 7$	$- 1$
		$- 2$
	$8$

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.

$- \frac{1}{4}$	$8$	$- 10$	$- 7$	$- 1$
		$- 2$
	$8$	$- 12$

Step 6.5

Multiply the newest entry in the result $(- 12)$ by the divisor $(- \frac{1}{4})$ and place the result of $(3)$ under the next term in the dividend $(- 7)$ .

$- \frac{1}{4}$	$8$	$- 10$	$- 7$	$- 1$
		$- 2$	$3$
	$8$	$- 12$

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.

$- \frac{1}{4}$	$8$	$- 10$	$- 7$	$- 1$
		$- 2$	$3$
	$8$	$- 12$	$- 4$

Step 6.7

Multiply the newest entry in the result $(- 4)$ by the divisor $(- \frac{1}{4})$ and place the result of $(1)$ under the next term in the dividend $(- 1)$ .

$- \frac{1}{4}$	$8$	$- 10$	$- 7$	$- 1$
		$- 2$	$3$	$1$
	$8$	$- 12$	$- 4$

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.

$- \frac{1}{4}$	$8$	$- 10$	$- 7$	$- 1$
		$- 2$	$3$	$1$
	$8$	$- 12$	$- 4$	$0$

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.

$8 x^{2} + - 12 x - 4$

Step 6.10

Simplify the quotient polynomial.

$8 x^{2} - 12 x - 4$

Step 7

Factor $4$ out of $8 x^{2} - 12 x - 4$ .

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Step 7.1

Factor $4$ out of $8 x^{2}$ .

$(x + \frac{1}{4}) (4 (2 x^{2}) - 12 x - 4)$

Step 7.2

Factor $4$ out of $- 12 x$ .

$(x + \frac{1}{4}) (4 (2 x^{2}) + 4 (- 3 x) - 4)$

Step 7.3

Factor $4$ out of $- 4$ .

$(x + \frac{1}{4}) (4 (2 x^{2}) + 4 (- 3 x) + 4 (- 1))$

Step 7.4

Factor $4$ out of $4 (2 x^{2}) + 4 (- 3 x)$ .

$(x + \frac{1}{4}) (4 (2 x^{2} - 3 x) + 4 (- 1))$

Step 7.5

Factor $4$ out of $4 (2 x^{2} - 3 x) + 4 (- 1)$ .

$(x + \frac{1}{4}) (4 (2 x^{2} - 3 x - 1))$

Step 8

Factor $8 x^{3} - 10 x^{2} - 7 x - 1$ 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.

$p = \pm 1$

$q = \pm 1, \pm 8, \pm 2, \pm 4$

Step 8.2

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

$\pm 1, \pm 0.125, \pm 0.5, \pm 0.25$

Step 8.3

Substitute $- 0.25$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.25$ is a root of the polynomial.

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Step 8.3.1

Substitute $- 0.25$ into the polynomial.

$8 {(- 0.25)}^{3} - 10 {(- 0.25)}^{2} - 7 \cdot - 0.25 - 1$

Step 8.3.2

Raise $- 0.25$ to the power of $3$ .

$8 \cdot - 0.015625 - 10 {(- 0.25)}^{2} - 7 \cdot - 0.25 - 1$

Step 8.3.3

Multiply $8$ by $- 0.015625$ .

$- 0.125 - 10 {(- 0.25)}^{2} - 7 \cdot - 0.25 - 1$

Step 8.3.4

Raise $- 0.25$ to the power of $2$ .

$- 0.125 - 10 \cdot 0.0625 - 7 \cdot - 0.25 - 1$

Step 8.3.5

Multiply $- 10$ by $0.0625$ .

$- 0.125 - 0.625 - 7 \cdot - 0.25 - 1$

Step 8.3.6

Subtract $0.625$ from $- 0.125$ .

$- 0.75 - 7 \cdot - 0.25 - 1$

Step 8.3.7

Multiply $- 7$ by $- 0.25$ .

$- 0.75 + 1.75 - 1$

Step 8.3.8

Add $- 0.75$ and $1.75$ .

$1 - 1$

Step 8.3.9

Subtract $1$ from $1$ .

$0$

Step 8.4

Since $- 0.25$ is a known root, divide the polynomial by $4 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{8 x^{3} - 10 x^{2} - 7 x - 1}{4 x + 1}$

Step 8.5

Divide $8 x^{3} - 10 x^{2} - 7 x - 1$ by $4 x + 1$ .

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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 $0$ .

$4 x$

$1$

$8 x^{3}$

$10 x^{2}$

$7 x$

$1$

Step 8.5.2

Divide the highest order term in the dividend $8 x^{3}$ by the highest order term in divisor $4 x$ .

					$2 x^{2}$
	$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$

Step 8.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			+	$8 x^{3}$	+	$2 x^{2}$

Step 8.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{3} + 2 x^{2}$

				$2 x^{2}$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$

Step 8.5.5

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

				$2 x^{2}$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$

Step 8.5.6

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

				$2 x^{2}$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$

Step 8.5.7

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $4 x$ .

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$

Step 8.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					-	$12 x^{2}$	-	$3 x$

Step 8.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} - 3 x$

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$

Step 8.5.10

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

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$

Step 8.5.11

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

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$

Step 8.5.12

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $4 x$ .

				$2 x^{2}$	-	$3 x$	-	$1$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$

Step 8.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$	-	$1$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$
							-	$4 x$	-	$1$

Step 8.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 1$

				$2 x^{2}$	-	$3 x$	-	$1$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$
							+	$4 x$	+	$1$

Step 8.5.15

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

				$2 x^{2}$	-	$3 x$	-	$1$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$
							+	$4 x$	+	$1$
										$0$

Step 8.5.16

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

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

Step 8.6

Write $8 x^{3} - 10 x^{2} - 7 x - 1$ as a set of factors.

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

Step 9

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

$4 x + 1 = 0$

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

Step 10

Set $4 x + 1$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Step 10.2

Solve $4 x + 1 = 0$ for $x$ .

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Step 10.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 10.2.2

Divide each term in $4 x = - 1$ by $4$ and simplify.

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Step 10.2.2.1

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $4$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

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.

$x = - \frac{1}{4}$

Step 11

Set $2 x^{2} - 3 x - 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $2 x^{2} - 3 x - 1$ equal to $0$ .

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

Step 11.2

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

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

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 11.2.3.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 11.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 11.2.3.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 11.2.3.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 2}$

Step 11.2.3.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{17}}{4}$

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 11.2.4.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 11.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 11.2.4.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 11.2.4.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 2}$

Step 11.2.4.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{17}}{4}$

Step 11.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{17}}{4}$

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 11.2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 11.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 11.2.5.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 11.2.5.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 2}$

Step 11.2.5.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{17}}{4}$

Step 11.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{17}}{4}$

Step 11.2.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Step 12

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

$x = - \frac{1}{4}, \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{4}, \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Decimal Form:

$x = - 0.25, 1.78077640 \dots, - 0.28077640 \dots$

Step 14