Algebra Examples

$\sqrt[3]{x + 1} = 2 x + 2$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{x + 1}}^{3} = {(2 x + 2)}^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 1}$ as ${(x + 1)}^{\frac{1}{3}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x + 1)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x + 1)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x + 1)}^{\frac{1}{3} \cdot 3} = {(2 x + 2)}^{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x + 1)}^{\frac{1}{3} \cdot 3} = {(2 x + 2)}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(2 x + 2)}^{3}$ .

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Step 2.3.1.1

Use the Binomial Theorem.

$x + 1 = {(2 x)}^{3} + 3 {(2 x)}^{2} \cdot 2 + 3 (2 x) \cdot 2^{2} + 2^{3}$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

Apply the product rule to $2 x$ .

$x + 1 = 2^{3} x^{3} + 3 {(2 x)}^{2} \cdot 2 + 3 (2 x) \cdot 2^{2} + 2^{3}$

Step 2.3.1.2.2

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

$x + 1 = 8 x^{3} + 3 {(2 x)}^{2} \cdot 2 + 3 (2 x) \cdot 2^{2} + 2^{3}$

Step 2.3.1.2.3

Apply the product rule to $2 x$ .

$x + 1 = 8 x^{3} + 3 (2^{2} x^{2}) \cdot 2 + 3 (2 x) \cdot 2^{2} + 2^{3}$

Step 2.3.1.2.4

Multiply $2^{2}$ by $2$ by adding the exponents.

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Step 2.3.1.2.4.1

Move $2$ .

$x + 1 = 8 x^{3} + 3 (2 \cdot 2^{2} x^{2}) + 3 (2 x) \cdot 2^{2} + 2^{3}$

Step 2.3.1.2.4.2

Multiply $2$ by $2^{2}$ .

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Step 2.3.1.2.4.2.1

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

$x + 1 = 8 x^{3} + 3 (2^{1} \cdot 2^{2} x^{2}) + 3 (2 x) \cdot 2^{2} + 2^{3}$

Step 2.3.1.2.4.2.2

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

$x + 1 = 8 x^{3} + 3 (2^{1 + 2} x^{2}) + 3 (2 x) \cdot 2^{2} + 2^{3}$

Step 2.3.1.2.4.3

Add $1$ and $2$ .

$x + 1 = 8 x^{3} + 3 (2^{3} x^{2}) + 3 (2 x) \cdot 2^{2} + 2^{3}$

Step 2.3.1.2.5

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

$x + 1 = 8 x^{3} + 3 (8 x^{2}) + 3 (2 x) \cdot 2^{2} + 2^{3}$

Step 2.3.1.2.6

Multiply $8$ by $3$ .

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

Step 2.3.1.2.7

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 2.3.1.2.7.1

Move $2^{2}$ .

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

Step 2.3.1.2.7.2

Multiply $2^{2}$ by $2$ .

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Step 2.3.1.2.7.2.1

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

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

Step 2.3.1.2.7.2.2

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

$x + 1 = 8 x^{3} + 24 x^{2} + 3 (2^{2 + 1} x) + 2^{3}$

Step 2.3.1.2.7.3

Add $2$ and $1$ .

$x + 1 = 8 x^{3} + 24 x^{2} + 3 (2^{3} x) + 2^{3}$

Step 2.3.1.2.8

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

$x + 1 = 8 x^{3} + 24 x^{2} + 3 (8 x) + 2^{3}$

Step 2.3.1.2.9

Multiply $8$ by $3$ .

$x + 1 = 8 x^{3} + 24 x^{2} + 24 x + 2^{3}$

Step 2.3.1.2.10

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

$x + 1 = 8 x^{3} + 24 x^{2} + 24 x + 8$

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$8 x^{3} + 24 x^{2} + 24 x + 8 = x + 1$

Step 3.2

Move all terms containing $x$ to the left side of the equation.

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Step 3.2.1

Subtract $x$ from both sides of the equation.

$8 x^{3} + 24 x^{2} + 24 x + 8 - x = 1$

Step 3.2.2

Subtract $x$ from $24 x$ .

$8 x^{3} + 24 x^{2} + 23 x + 8 = 1$

Step 3.3

Subtract $1$ from both sides of the equation.

$8 x^{3} + 24 x^{2} + 23 x + 8 - 1 = 0$

Step 3.4

Subtract $1$ from $8$ .

$8 x^{3} + 24 x^{2} + 23 x + 7 = 0$

Step 3.5

Factor $8 x^{3} + 24 x^{2} + 23 x + 7$ using the rational roots test.

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Step 3.5.1

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.

$p = \pm 1, \pm 7$

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

Step 3.5.2

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

$\pm 1, \pm 0.125, \pm 0.5, \pm 0.25, \pm 7, \pm 0.875, \pm 3.5, \pm 1.75$

Step 3.5.3

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 3.5.3.1

Substitute $- 1$ into the polynomial.

$8 {(- 1)}^{3} + 24 {(- 1)}^{2} + 23 \cdot - 1 + 7$

Step 3.5.3.2

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

$8 \cdot - 1 + 24 {(- 1)}^{2} + 23 \cdot - 1 + 7$

Step 3.5.3.3

Multiply $8$ by $- 1$ .

$- 8 + 24 {(- 1)}^{2} + 23 \cdot - 1 + 7$

Step 3.5.3.4

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

$- 8 + 24 \cdot 1 + 23 \cdot - 1 + 7$

Step 3.5.3.5

Multiply $24$ by $1$ .

$- 8 + 24 + 23 \cdot - 1 + 7$

Step 3.5.3.6

Add $- 8$ and $24$ .

$16 + 23 \cdot - 1 + 7$

Step 3.5.3.7

Multiply $23$ by $- 1$ .

$16 - 23 + 7$

Step 3.5.3.8

Subtract $23$ from $16$ .

$- 7 + 7$

Step 3.5.3.9

Add $- 7$ and $7$ .

$0$

Step 3.5.4

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.

$\frac{8 x^{3} + 24 x^{2} + 23 x + 7}{x + 1}$

Step 3.5.5

Divide $8 x^{3} + 24 x^{2} + 23 x + 7$ by $x + 1$ .

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Step 3.5.5.1

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

$x$

$1$

$8 x^{3}$

$24 x^{2}$

$23 x$

$7$

Step 3.5.5.2

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

					$8 x^{2}$
	$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$

Step 3.5.5.3

Multiply the new quotient term by the divisor.

				$8 x^{2}$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			+	$8 x^{3}$	+	$8 x^{2}$

Step 3.5.5.4

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

				$8 x^{2}$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$

Step 3.5.5.5

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

				$8 x^{2}$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$

Step 3.5.5.6

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

				$8 x^{2}$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$

Step 3.5.5.7

Divide the highest order term in the dividend $16 x^{2}$ by the highest order term in divisor $x$ .

				$8 x^{2}$	+	$16 x$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$

Step 3.5.5.8

Multiply the new quotient term by the divisor.

				$8 x^{2}$	+	$16 x$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$
					+	$16 x^{2}$	+	$16 x$

Step 3.5.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{2} + 16 x$

				$8 x^{2}$	+	$16 x$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$
					-	$16 x^{2}$	-	$16 x$

Step 3.5.5.10

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

				$8 x^{2}$	+	$16 x$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$
					-	$16 x^{2}$	-	$16 x$
							+	$7 x$

Step 3.5.5.11

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

				$8 x^{2}$	+	$16 x$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$
					-	$16 x^{2}$	-	$16 x$
							+	$7 x$	+	$7$

Step 3.5.5.12

Divide the highest order term in the dividend $7 x$ by the highest order term in divisor $x$ .

				$8 x^{2}$	+	$16 x$	+	$7$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$
					-	$16 x^{2}$	-	$16 x$
							+	$7 x$	+	$7$

Step 3.5.5.13

Multiply the new quotient term by the divisor.

				$8 x^{2}$	+	$16 x$	+	$7$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$
					-	$16 x^{2}$	-	$16 x$
							+	$7 x$	+	$7$
							+	$7 x$	+	$7$

Step 3.5.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $7 x + 7$

				$8 x^{2}$	+	$16 x$	+	$7$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$
					-	$16 x^{2}$	-	$16 x$
							+	$7 x$	+	$7$
							-	$7 x$	-	$7$

Step 3.5.5.15

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

				$8 x^{2}$	+	$16 x$	+	$7$
$x$	+	$1$		$8 x^{3}$	+	$24 x^{2}$	+	$23 x$	+	$7$
			-	$8 x^{3}$	-	$8 x^{2}$
					+	$16 x^{2}$	+	$23 x$
					-	$16 x^{2}$	-	$16 x$
							+	$7 x$	+	$7$
							-	$7 x$	-	$7$
										$0$

Step 3.5.5.16

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

$8 x^{2} + 16 x + 7$

Step 3.5.6

Write $8 x^{3} + 24 x^{2} + 23 x + 7$ as a set of factors.

$(x + 1) (8 x^{2} + 16 x + 7) = 0$

Step 3.6

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

$x + 1 = 0$

$8 x^{2} + 16 x + 7 = 0$

Step 3.7

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

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Step 3.7.1

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

$x + 1 = 0$

Step 3.7.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.8

Set $8 x^{2} + 16 x + 7$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $8 x^{2} + 16 x + 7$ equal to $0$ .

$8 x^{2} + 16 x + 7 = 0$

Step 3.8.2

Solve $8 x^{2} + 16 x + 7 = 0$ for $x$ .

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Step 3.8.2.1

Use the quadratic formula to find the solutions.

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

Step 3.8.2.2

Substitute the values $a = 8$ , $b = 16$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (8 \cdot 7)}}{2 \cdot 8}$

Step 3.8.2.3

Simplify.

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Step 3.8.2.3.1

Simplify the numerator.

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Step 3.8.2.3.1.1

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 8 \cdot 7}}{2 \cdot 8}$

Step 3.8.2.3.1.2

Multiply $- 4 \cdot 8 \cdot 7$ .

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Step 3.8.2.3.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{- 16 \pm \sqrt{256 - 32 \cdot 7}}{2 \cdot 8}$

Step 3.8.2.3.1.2.2

Multiply $- 32$ by $7$ .

$x = \frac{- 16 \pm \sqrt{256 - 224}}{2 \cdot 8}$

Step 3.8.2.3.1.3

Subtract $224$ from $256$ .

$x = \frac{- 16 \pm \sqrt{32}}{2 \cdot 8}$

Step 3.8.2.3.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.8.2.3.1.4.1

Factor $16$ out of $32$ .

$x = \frac{- 16 \pm \sqrt{16 (2)}}{2 \cdot 8}$

Step 3.8.2.3.1.4.2

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

$x = \frac{- 16 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 8}$

Step 3.8.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 16 \pm 4 \sqrt{2}}{2 \cdot 8}$

Step 3.8.2.3.2

Multiply $2$ by $8$ .

$x = \frac{- 16 \pm 4 \sqrt{2}}{16}$

Step 3.8.2.3.3

Simplify $\frac{- 16 \pm 4 \sqrt{2}}{16}$ .

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

Step 3.8.2.4

The final answer is the combination of both solutions.

$x = - \frac{4 - \sqrt{2}}{4}, - \frac{4 + \sqrt{2}}{4}$

Step 3.9

The final solution is all the values that make $(x + 1) (8 x^{2} + 16 x + 7) = 0$ true.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1, - 0.64644660 \dots, - 1.35355339 \dots$