Algebra Examples

$2 x^{3} + 2 x - 3 = - 0.5 | x - 4 |$

Step 1

Rewrite the equation as $- 0.5 | x - 4 | = 2 x^{3} + 2 x - 3$ .

$- 0.5 | x - 4 | = 2 x^{3} + 2 x - 3$

Step 2

Divide each term in $- 0.5 | x - 4 | = 2 x^{3} + 2 x - 3$ by $- 0.5$ and simplify.

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Step 2.1

Divide each term in $- 0.5 | x - 4 | = 2 x^{3} + 2 x - 3$ by $- 0.5$ .

$\frac{- 0.5 | x - 4 |}{- 0.5} = \frac{2 x^{3}}{- 0.5} + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 0.5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 0.5 | x - 4 |}{- 0.5} = \frac{2 x^{3}}{- 0.5} + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.2.1.2

Divide $| x - 4 |$ by $1$ .

$| x - 4 | = \frac{2 x^{3}}{- 0.5} + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Move the negative in front of the fraction.

$| x - 4 | = - \frac{2 x^{3}}{0.5} + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.3.1.2

Factor $2$ out of $2 x^{3}$ .

$| x - 4 | = - \frac{2 (x^{3})}{0.5} + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.3.1.3

Factor $0.5$ out of $0.5$ .

$| x - 4 | = - \frac{2 (x^{3})}{0.5 (1)} + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.3.1.4

Separate fractions.

$| x - 4 | = - (\frac{2}{0.5} \cdot \frac{x^{3}}{1}) + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.3.1.5

Divide $2$ by $0.5$ .

$| x - 4 | = - (4 \frac{x^{3}}{1}) + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.3.1.6

Divide $x^{3}$ by $1$ .

$| x - 4 | = - (4 x^{3}) + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.3.1.7

Multiply $4$ by $- 1$ .

$| x - 4 | = - 4 x^{3} + \frac{2 x}{- 0.5} + \frac{- 3}{- 0.5}$

Step 2.3.1.8

Move the negative in front of the fraction.

$| x - 4 | = - 4 x^{3} - \frac{2 x}{0.5} + \frac{- 3}{- 0.5}$

Step 2.3.1.9

Factor $2$ out of $2 x$ .

$| x - 4 | = - 4 x^{3} - \frac{2 (x)}{0.5} + \frac{- 3}{- 0.5}$

Step 2.3.1.10

Factor $0.5$ out of $0.5$ .

$| x - 4 | = - 4 x^{3} - \frac{2 (x)}{0.5 (1)} + \frac{- 3}{- 0.5}$

Step 2.3.1.11

Separate fractions.

$| x - 4 | = - 4 x^{3} - (\frac{2}{0.5} \cdot \frac{x}{1}) + \frac{- 3}{- 0.5}$

Step 2.3.1.12

Divide $2$ by $0.5$ .

$| x - 4 | = - 4 x^{3} - (4 \frac{x}{1}) + \frac{- 3}{- 0.5}$

Step 2.3.1.13

Divide $x$ by $1$ .

$| x - 4 | = - 4 x^{3} - (4 x) + \frac{- 3}{- 0.5}$

Step 2.3.1.14

Multiply $4$ by $- 1$ .

$| x - 4 | = - 4 x^{3} - 4 x + \frac{- 3}{- 0.5}$

Step 2.3.1.15

Divide $- 3$ by $- 0.5$ .

$| x - 4 | = - 4 x^{3} - 4 x + 6$

Step 3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 4 = \pm (- 4 x^{3} - 4 x + 6)$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 4 = - 4 x^{3} - 4 x + 6$

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x - 4 = - (- 4 x^{3} - 4 x + 6)$

Step 4.3

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

$- (- 4 x^{3} - 4 x + 6) = x - 4$

Step 4.4

Simplify $- (- 4 x^{3} - 4 x + 6)$ .

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Step 4.4.1

Rewrite.

$0 + 0 - (- 4 x^{3} - 4 x + 6) = x - 4$

Step 4.4.2

Simplify by adding zeros.

$- (- 4 x^{3} - 4 x + 6) = x - 4$

Step 4.4.3

Apply the distributive property.

$- (- 4 x^{3}) - (- 4 x) - 1 \cdot 6 = x - 4$

Step 4.4.4

Simplify.

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Step 4.4.4.1

Multiply $- 4$ by $- 1$ .

$4 x^{3} - (- 4 x) - 1 \cdot 6 = x - 4$

Step 4.4.4.2

Multiply $- 4$ by $- 1$ .

$4 x^{3} + 4 x - 1 \cdot 6 = x - 4$

Step 4.4.4.3

Multiply $- 1$ by $6$ .

$4 x^{3} + 4 x - 6 = x - 4$

Step 4.5

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

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Step 4.5.1

Subtract $x$ from both sides of the equation.

$4 x^{3} + 4 x - 6 - x = - 4$

Step 4.5.2

Subtract $x$ from $4 x$ .

$4 x^{3} + 3 x - 6 = - 4$

Step 4.6

Add $4$ to both sides of the equation.

$4 x^{3} + 3 x - 6 + 4 = 0$

Step 4.7

Add $- 6$ and $4$ .

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

Step 4.8

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

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Step 4.8.1

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.

$p = \pm 1, \pm 2$

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

Step 4.8.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 2$

Step 4.8.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 4.8.3.1

Substitute $0.5$ into the polynomial.

$4 \cdot {0.5}^{3} + 3 \cdot 0.5 - 2$

Step 4.8.3.2

Raise $0.5$ to the power of $3$ .

$4 \cdot 0.125 + 3 \cdot 0.5 - 2$

Step 4.8.3.3

Multiply $4$ by $0.125$ .

$0.5 + 3 \cdot 0.5 - 2$

Step 4.8.3.4

Multiply $3$ by $0.5$ .

$0.5 + 1.5 - 2$

Step 4.8.3.5

Add $0.5$ and $1.5$ .

$2 - 2$

Step 4.8.3.6

Subtract $2$ from $2$ .

$0$

Step 4.8.4

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 4.8.5

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

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Step 4.8.5.1

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

$2 x$

$1$

$4 x^{3}$

$0 x^{2}$

$3 x$

$2$

Step 4.8.5.2

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

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

Step 4.8.5.3

Multiply the new quotient term by the divisor.

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

Step 4.8.5.4

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

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

Step 4.8.5.5

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

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

Step 4.8.5.6

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

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

Step 4.8.5.7

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

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

Step 4.8.5.8

Multiply the new quotient term by the divisor.

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

Step 4.8.5.9

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

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

Step 4.8.5.10

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

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

Step 4.8.5.11

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

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

Step 4.8.5.12

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

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

Step 4.8.5.13

Multiply the new quotient term by the divisor.

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

Step 4.8.5.14

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

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

Step 4.8.5.15

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

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

Step 4.8.5.16

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

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

Step 4.8.6

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

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

Step 4.9

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

$2 x - 1 = 0$

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

Step 4.10

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

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Step 4.10.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 4.10.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 4.10.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 4.10.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 4.10.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 4.10.2.2.2

Simplify the left side.

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Step 4.10.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.10.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 4.10.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 4.11

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

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Step 4.11.1

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

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

Step 4.11.2

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

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Step 4.11.2.1

Use the quadratic formula to find the solutions.

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

Step 4.11.2.2

Substitute the values $a = 2$ , $b = 1$ , and $c = 2$ into the quadratic formula and solve for $x$ .

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

Step 4.11.2.3

Simplify.

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Step 4.11.2.3.1

Simplify the numerator.

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Step 4.11.2.3.1.1

One to any power is one.

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

Step 4.11.2.3.1.2

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

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Step 4.11.2.3.1.2.1

Multiply $- 4$ by $2$ .

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

Step 4.11.2.3.1.2.2

Multiply $- 8$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

Step 4.11.2.3.1.3

Subtract $16$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 15}}{2 \cdot 2}$

Step 4.11.2.3.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

Step 4.11.2.3.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

Step 4.11.2.3.1.6

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

$x = \frac{- 1 \pm i \sqrt{15}}{2 \cdot 2}$

Step 4.11.2.3.2

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm i \sqrt{15}}{4}$

Step 4.11.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}$

Step 4.12

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

$x = \frac{1}{2}, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}$