Algebra Examples

$x^{4} - 4 x^{3} = 6 x^{2} - 12 x$

Step 1

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

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Step 1.1

Subtract $6 x^{2}$ from both sides of the equation.

$x^{4} - 4 x^{3} - 6 x^{2} = - 12 x$

Step 1.2

Add $12 x$ to both sides of the equation.

$x^{4} - 4 x^{3} - 6 x^{2} + 12 x = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $x$ out of $x^{4} - 4 x^{3} - 6 x^{2} + 12 x$ .

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Step 2.1.1

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

$x \cdot x^{3} - 4 x^{3} - 6 x^{2} + 12 x = 0$

Step 2.1.2

Factor $x$ out of $- 4 x^{3}$ .

$x \cdot x^{3} + x (- 4 x^{2}) - 6 x^{2} + 12 x = 0$

Step 2.1.3

Factor $x$ out of $- 6 x^{2}$ .

$x \cdot x^{3} + x (- 4 x^{2}) + x (- 6 x) + 12 x = 0$

Step 2.1.4

Factor $x$ out of $12 x$ .

$x \cdot x^{3} + x (- 4 x^{2}) + x (- 6 x) + x \cdot 12 = 0$

Step 2.1.5

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

$x (x^{3} - 4 x^{2}) + x (- 6 x) + x \cdot 12 = 0$

Step 2.1.6

Factor $x$ out of $x (x^{3} - 4 x^{2}) + x (- 6 x)$ .

$x (x^{3} - 4 x^{2} - 6 x) + x \cdot 12 = 0$

Step 2.1.7

Factor $x$ out of $x (x^{3} - 4 x^{2} - 6 x) + x \cdot 12$ .

$x (x^{3} - 4 x^{2} - 6 x + 12) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $x^{3} - 4 x^{2} - 6 x + 12$ using the rational roots test.

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Step 2.2.1.1

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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1$

Step 2.2.1.2

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Step 2.2.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 2.2.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - 4 {(- 2)}^{2} - 6 \cdot - 2 + 12$

Step 2.2.1.3.2

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

$- 8 - 4 {(- 2)}^{2} - 6 \cdot - 2 + 12$

Step 2.2.1.3.3

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

$- 8 - 4 \cdot 4 - 6 \cdot - 2 + 12$

Step 2.2.1.3.4

Multiply $- 4$ by $4$ .

$- 8 - 16 - 6 \cdot - 2 + 12$

Step 2.2.1.3.5

Subtract $16$ from $- 8$ .

$- 24 - 6 \cdot - 2 + 12$

Step 2.2.1.3.6

Multiply $- 6$ by $- 2$ .

$- 24 + 12 + 12$

Step 2.2.1.3.7

Add $- 24$ and $12$ .

$- 12 + 12$

Step 2.2.1.3.8

Add $- 12$ and $12$ .

$0$

Step 2.2.1.4

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

$\frac{x^{3} - 4 x^{2} - 6 x + 12}{x + 2}$

Step 2.2.1.5

Divide $x^{3} - 4 x^{2} - 6 x + 12$ by $x + 2$ .

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Step 2.2.1.5.1

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

$x$

$2$

$x^{3}$

$4 x^{2}$

$6 x$

$12$

Step 2.2.1.5.2

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$

Step 2.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			+	$x^{3}$	+	$2 x^{2}$

Step 2.2.1.5.4

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$

Step 2.2.1.5.5

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$

Step 2.2.1.5.6

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$

Step 2.2.1.5.7

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$

Step 2.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$
					-	$6 x^{2}$	-	$12 x$

Step 2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 12 x$

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$
					+	$6 x^{2}$	+	$12 x$

Step 2.2.1.5.10

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$
					+	$6 x^{2}$	+	$12 x$
							+	$6 x$

Step 2.2.1.5.11

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$
					+	$6 x^{2}$	+	$12 x$
							+	$6 x$	+	$12$

Step 2.2.1.5.12

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

				$x^{2}$	-	$6 x$	+	$6$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$
					+	$6 x^{2}$	+	$12 x$
							+	$6 x$	+	$12$

Step 2.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	+	$6$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$
					+	$6 x^{2}$	+	$12 x$
							+	$6 x$	+	$12$
							+	$6 x$	+	$12$

Step 2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + 12$

				$x^{2}$	-	$6 x$	+	$6$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$
					+	$6 x^{2}$	+	$12 x$
							+	$6 x$	+	$12$
							-	$6 x$	-	$12$

Step 2.2.1.5.15

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

				$x^{2}$	-	$6 x$	+	$6$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$6 x$	+	$12$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$6 x$
					+	$6 x^{2}$	+	$12 x$
							+	$6 x$	+	$12$
							-	$6 x$	-	$12$
										$0$

Step 2.2.1.5.16

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

$x^{2} - 6 x + 6$

Step 2.2.1.6

Write $x^{3} - 4 x^{2} - 6 x + 12$ as a set of factors.

$x ((x + 2) (x^{2} - 6 x + 6)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$x (x + 2) (x^{2} - 6 x + 6) = 0$

Step 3

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

$x = 0$

$x + 2 = 0$

$x^{2} - 6 x + 6 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

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

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Step 5.1

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

$x + 2 = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

Set $x^{2} - 6 x + 6$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{2} - 6 x + 6$ equal to $0$ .

$x^{2} - 6 x + 6 = 0$

Step 6.2

Solve $x^{2} - 6 x + 6 = 0$ for $x$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

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

Step 6.2.2

Substitute the values $a = 1$ , $b = - 6$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 6)}}{2 \cdot 1}$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 6}}{2 \cdot 1}$

Step 6.2.3.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{6 \pm \sqrt{36 - 24}}{2 \cdot 1}$

Step 6.2.3.1.3

Subtract $24$ from $36$ .

$x = \frac{6 \pm \sqrt{12}}{2 \cdot 1}$

Step 6.2.3.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.2.3.1.4.1

Factor $4$ out of $12$ .

$x = \frac{6 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 6.2.3.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 6.2.3.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{3}}{2}$

Step 6.2.3.3

Simplify $\frac{6 \pm 2 \sqrt{3}}{2}$ .

$x = 3 \pm \sqrt{3}$

Step 6.2.4

The final answer is the combination of both solutions.

$x = 3 + \sqrt{3}, 3 - \sqrt{3}$

Step 7

The final solution is all the values that make $x (x + 2) (x^{2} - 6 x + 6) = 0$ true.

$x = 0, - 2, 3 + \sqrt{3}, 3 - \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, - 2, 3 + \sqrt{3}, 3 - \sqrt{3}$

Decimal Form:

$x = 0, - 2, 4.73205080 \dots, 1.26794919 \dots$

Step 9