Precalculus Examples

$48 x^{3} - 80 x^{2} + 41 x - 6 = 0$ , $x = \frac{2}{3}$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $48 x^{3} - 80 x^{2} + 41 x - 6$ using the rational roots test.

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Step 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.

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

$q = \pm 1, \pm 48, \pm 2, \pm 24, \pm 3, \pm 16, \pm 4, \pm 12, \pm 6, \pm 8$

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

Step 1.1.2

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

$\pm 1, \pm 0.0208\overline{3}, \pm 0.5, \pm 0.041\overline{6}, \pm 0.\overline{3}, \pm 0.0625, \pm 0.25, \pm 0.08\overline{3}, \pm 0.1\overline{6}, \pm 0.125, \pm 6, \pm 3, \pm 2, \pm 0.375, \pm 1.5, \pm 0.75, \pm 0.\overline{6}, \pm 0.1875$

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

Step 1.1.3

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 1.1.3.1

Substitute $0.25$ into the polynomial.

$48 \cdot {0.25}^{3} - 80 \cdot {0.25}^{2} + 41 \cdot 0.25 - 6$

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

Step 1.1.3.2

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

$48 \cdot 0.015625 - 80 \cdot {0.25}^{2} + 41 \cdot 0.25 - 6$

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

Step 1.1.3.3

Multiply $48$ by $0.015625$ .

$0.75 - 80 \cdot {0.25}^{2} + 41 \cdot 0.25 - 6$

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

Step 1.1.3.4

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

$0.75 - 80 \cdot 0.0625 + 41 \cdot 0.25 - 6$

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

Step 1.1.3.5

Multiply $- 80$ by $0.0625$ .

$0.75 - 5 + 41 \cdot 0.25 - 6$

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

Step 1.1.3.6

Subtract $5$ from $0.75$ .

$- 4.25 + 41 \cdot 0.25 - 6$

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

Step 1.1.3.7

Multiply $41$ by $0.25$ .

$- 4.25 + 10.25 - 6$

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

Step 1.1.3.8

Add $- 4.25$ and $10.25$ .

$6 - 6$

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

Step 1.1.3.9

Subtract $6$ from $6$ .

$0$

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

$0$

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

Step 1.1.4

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.

$\frac{48 x^{3} - 80 x^{2} + 41 x - 6}{4 x - 1}$

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

Step 1.1.5

Divide $48 x^{3} - 80 x^{2} + 41 x - 6$ by $4 x - 1$ .

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Step 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 $0$ .

$4 x$

$1$

$48 x^{3}$

$80 x^{2}$

$41 x$

$6$

Step 1.1.5.2

Divide the highest order term in the dividend $48 x^{3}$ by the highest order term in divisor $4 x$ .

					$12 x^{2}$
	$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

				$12 x^{2}$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			+	$48 x^{3}$	-	$12 x^{2}$

Step 1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $48 x^{3} - 12 x^{2}$

				$12 x^{2}$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$

Step 1.1.5.5

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

				$12 x^{2}$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$

Step 1.1.5.6

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

				$12 x^{2}$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$

Step 1.1.5.7

Divide the highest order term in the dividend $- 68 x^{2}$ by the highest order term in divisor $4 x$ .

				$12 x^{2}$	-	$17 x$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

				$12 x^{2}$	-	$17 x$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$
					-	$68 x^{2}$	+	$17 x$

Step 1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 68 x^{2} + 17 x$

				$12 x^{2}$	-	$17 x$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$
					+	$68 x^{2}$	-	$17 x$

Step 1.1.5.10

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

				$12 x^{2}$	-	$17 x$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$
					+	$68 x^{2}$	-	$17 x$
							+	$24 x$

Step 1.1.5.11

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

				$12 x^{2}$	-	$17 x$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$
					+	$68 x^{2}$	-	$17 x$
							+	$24 x$	-	$6$

Step 1.1.5.12

Divide the highest order term in the dividend $24 x$ by the highest order term in divisor $4 x$ .

				$12 x^{2}$	-	$17 x$	+	$6$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$
					+	$68 x^{2}$	-	$17 x$
							+	$24 x$	-	$6$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

				$12 x^{2}$	-	$17 x$	+	$6$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$
					+	$68 x^{2}$	-	$17 x$
							+	$24 x$	-	$6$
							+	$24 x$	-	$6$

Step 1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $24 x - 6$

				$12 x^{2}$	-	$17 x$	+	$6$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$
					+	$68 x^{2}$	-	$17 x$
							+	$24 x$	-	$6$
							-	$24 x$	+	$6$

Step 1.1.5.15

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

				$12 x^{2}$	-	$17 x$	+	$6$
$4 x$	-	$1$		$48 x^{3}$	-	$80 x^{2}$	+	$41 x$	-	$6$
			-	$48 x^{3}$	+	$12 x^{2}$
					-	$68 x^{2}$	+	$41 x$
					+	$68 x^{2}$	-	$17 x$
							+	$24 x$	-	$6$
							-	$24 x$	+	$6$
										$0$

Step 1.1.5.16

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

$12 x^{2} - 17 x + 6$

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

$12 x^{2} - 17 x + 6$

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

Step 1.1.6

Write $48 x^{3} - 80 x^{2} + 41 x - 6$ as a set of factors.

$(4 x - 1) (12 x^{2} - 17 x + 6) = 0$

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

$(4 x - 1) (12 x^{2} - 17 x + 6) = 0$

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

Step 1.2

Factor by grouping.

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Step 1.2.1

Factor by grouping.

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Step 1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 12 \cdot 6 = 72$ and whose sum is $b = - 17$ .

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Step 1.2.1.1.1

Factor $- 17$ out of $- 17 x$ .

$(4 x - 1) (12 x^{2} - 17 x + 6) = 0$

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

Step 1.2.1.1.2

Rewrite $- 17$ as $- 8$ plus $- 9$

$(4 x - 1) (12 x^{2} + (- 8 - 9) x + 6) = 0$

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

Step 1.2.1.1.3

Apply the distributive property.

$(4 x - 1) (12 x^{2} - 8 x - 9 x + 6) = 0$

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

$(4 x - 1) (12 x^{2} - 8 x - 9 x + 6) = 0$

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

Step 1.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.1.2.1

Group the first two terms and the last two terms.

$(4 x - 1) ((12 x^{2} - 8 x) - 9 x + 6) = 0$

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

Step 1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(4 x - 1) (4 x (3 x - 2) - 3 (3 x - 2)) = 0$

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

$(4 x - 1) (4 x (3 x - 2) - 3 (3 x - 2)) = 0$

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

Step 1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$(4 x - 1) ((3 x - 2) (4 x - 3)) = 0$

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

$(4 x - 1) ((3 x - 2) (4 x - 3)) = 0$

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

Step 1.2.2

Remove unnecessary parentheses.

$(4 x - 1) (3 x - 2) (4 x - 3) = 0$

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

$(4 x - 1) (3 x - 2) (4 x - 3) = 0$

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

$(4 x - 1) (3 x - 2) (4 x - 3) = 0$

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

Step 2

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

$4 x - 1 = 0$

$3 x - 2 = 0$

$4 x - 3 = 0$

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

Step 3

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

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Step 3.1

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

$4 x - 1 = 0$

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

Step 3.2

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

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Step 3.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

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

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

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

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

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

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

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

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

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

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

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

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

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

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

Step 4

Set $3 x - 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

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

Step 4.2

Solve $3 x - 2 = 0$ for $x$ .

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Step 4.2.1

Add $2$ to both sides of the equation.

$3 x = 2$

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

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

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

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

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

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

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

Step 5.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 5.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

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

Step 5.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 5.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

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

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

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

$x = \frac{3}{4}$

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

$x = \frac{3}{4}$

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

$x = \frac{3}{4}$

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

$x = \frac{3}{4}$

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

$x = \frac{3}{4}$

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

Step 6

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

$x = \frac{1}{4}, \frac{2}{3}, \frac{3}{4}$

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

Step 7

Since the roots of an equation are the points where the solution is $0$ , set each root as a factor of the equation that equals $0$ .

$(x - (\frac{1}{4}, \frac{2}{3}, \frac{3}{4})) (x - \frac{2}{3}) = 0$

Step 8

Simplify.

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Step 8.1

Simplify terms.

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Step 8.1.1

Apply the distributive property.

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x + (x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) (- \frac{2}{3}) = 0$

Step 8.1.2

Combine $(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4})$ and $\frac{2}{3}$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) \cdot 2}{3} = 0$

Step 8.2

Simplify each term.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(x \cdot \frac{4}{4} - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) \cdot 2}{3} = 0$

Step 8.2.1.2

Combine $x$ and $\frac{4}{4}$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(\frac{x \cdot 4}{4} - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) \cdot 2}{3} = 0$

Step 8.2.1.3

Combine the numerators over the common denominator.

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(\frac{x \cdot 4 - 1}{4}, \frac{2}{3}, \frac{3}{4}) \cdot 2}{3} = 0$

Step 8.2.1.4

Move $4$ to the left of $x$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(\frac{4 x - 1}{4}, \frac{2}{3}, \frac{3}{4}) \cdot 2}{3} = 0$

Step 8.2.2

Move $2$ to the left of $(\frac{4 x - 1}{4}, \frac{2}{3}, \frac{3}{4})$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{2 \cdot (\frac{4 x - 1}{4}, \frac{2}{3}, \frac{3}{4})}{3} = 0$

Step 8.2.3

Simplify the numerator.

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Step 8.2.3.1

Multiply $2$ by each element of the matrix.

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(2 (\frac{4 x - 1}{4}), 2 (\frac{2}{3}), 2 (\frac{3}{4}))}{3} = 0$

Step 8.2.3.2

Simplify each element in the matrix.

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Step 8.2.3.2.1

Cancel the common factor of $2$ .

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Step 8.2.3.2.1.1

Factor $2$ out of $4$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(2 (\frac{4 x - 1}{2 (2)}), 2 (\frac{2}{3}), 2 (\frac{3}{4}))}{3} = 0$

Step 8.2.3.2.1.2

Cancel the common factor.

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(2 (\frac{4 x - 1}{2 \cdot 2}), 2 (\frac{2}{3}), 2 (\frac{3}{4}))}{3} = 0$

Step 8.2.3.2.1.3

Rewrite the expression.

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(\frac{4 x - 1}{2}, 2 (\frac{2}{3}), 2 (\frac{3}{4}))}{3} = 0$

Step 8.2.3.2.2

Multiply $2 (\frac{2}{3})$ .

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Step 8.2.3.2.2.1

Combine $2$ and $\frac{2}{3}$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(\frac{4 x - 1}{2}, \frac{2 \cdot 2}{3}, 2 (\frac{3}{4}))}{3} = 0$

Step 8.2.3.2.2.2

Multiply $2$ by $2$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(\frac{4 x - 1}{2}, \frac{4}{3}, 2 (\frac{3}{4}))}{3} = 0$

Step 8.2.3.2.3

Cancel the common factor of $2$ .

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Step 8.2.3.2.3.1

Factor $2$ out of $4$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(\frac{4 x - 1}{2}, \frac{4}{3}, 2 (\frac{3}{2 (2)}))}{3} = 0$

Step 8.2.3.2.3.2

Cancel the common factor.

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(\frac{4 x - 1}{2}, \frac{4}{3}, 2 (\frac{3}{2 \cdot 2}))}{3} = 0$

Step 8.2.3.2.3.3

Rewrite the expression.

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x - \frac{(\frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.3

To write $(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x \cdot \frac{3}{3} - \frac{(\frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.4

Simplify terms.

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Step 8.4.1

Combine $(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x$ and $\frac{3}{3}$ .

$\frac{(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x \cdot 3}{3} - \frac{(\frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.4.2

Combine the numerators over the common denominator.

$\frac{(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x \cdot 3 - (\frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Move $3$ to the left of $(x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x$ .

$\frac{(3 \cdot ((x - \frac{1}{4}, \frac{2}{3}, \frac{3}{4}) x) - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.2

Multiply $3$ by each element of the matrix.

$\frac{((3 (x - \frac{1}{4}), 3 (\frac{2}{3}), 3 (\frac{3}{4})) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.3

Simplify each element in the matrix.

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Step 8.5.3.1

Apply the distributive property.

$\frac{((3 x + 3 (- \frac{1}{4}), 3 (\frac{2}{3}), 3 (\frac{3}{4})) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.3.2

Multiply $3 (- \frac{1}{4})$ .

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Step 8.5.3.2.1

Multiply $- 1$ by $3$ .

$\frac{((3 x - 3 (\frac{1}{4}), 3 (\frac{2}{3}), 3 (\frac{3}{4})) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.3.2.2

Combine $- 3$ and $\frac{1}{4}$ .

$\frac{((3 x + \frac{- 3}{4}, 3 (\frac{2}{3}), 3 (\frac{3}{4})) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.3.3

Move the negative in front of the fraction.

$\frac{((3 x - \frac{3}{4}, 3 (\frac{2}{3}), 3 (\frac{3}{4})) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.3.4

Cancel the common factor of $3$ .

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Step 8.5.3.4.1

Cancel the common factor.

$\frac{((3 x - \frac{3}{4}, 3 (\frac{2}{3}), 3 (\frac{3}{4})) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.3.4.2

Rewrite the expression.

$\frac{((3 x - \frac{3}{4}, 2, 3 (\frac{3}{4})) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.3.5

Multiply $3 (\frac{3}{4})$ .

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Step 8.5.3.5.1

Combine $3$ and $\frac{3}{4}$ .

$\frac{((3 x - \frac{3}{4}, 2, \frac{3 \cdot 3}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.3.5.2

Multiply $3$ by $3$ .

$\frac{((3 x - \frac{3}{4}, 2, \frac{9}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.4

Simplify each term.

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Step 8.5.4.1

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{((3 x \cdot \frac{4}{4} - \frac{3}{4}, 2, \frac{9}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.4.2

Combine $3 x$ and $\frac{4}{4}$ .

$\frac{((\frac{3 x \cdot 4}{4} - \frac{3}{4}, 2, \frac{9}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.4.3

Combine the numerators over the common denominator.

$\frac{((\frac{3 x \cdot 4 - 3}{4}, 2, \frac{9}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.4.4

Simplify the numerator.

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Step 8.5.4.4.1

Factor $3$ out of $3 x \cdot 4 - 3$ .

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Step 8.5.4.4.1.1

Factor $3$ out of $3 x \cdot 4$ .

$\frac{((\frac{3 (x \cdot 4) - 3}{4}, 2, \frac{9}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.4.4.1.2

Factor $3$ out of $- 3$ .

$\frac{((\frac{3 (x \cdot 4) + 3 (- 1)}{4}, 2, \frac{9}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.4.4.1.3

Factor $3$ out of $3 (x \cdot 4) + 3 (- 1)$ .

$\frac{((\frac{3 (x \cdot 4 - 1)}{4}, 2, \frac{9}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.5.4.4.2

Move $4$ to the left of $x$ .

$\frac{((\frac{3 (4 x - 1)}{4}, 2, \frac{9}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$

Step 8.6

Reorder factors in $\frac{(\frac{3 (4 x - 1)}{4}, 2, \frac{9}{4}) x - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2}}{3}$ .

$\frac{(x (\frac{3 (4 x - 1)}{4}, 2, \frac{9}{4}) - \frac{4 x - 1}{2}, \frac{4}{3}, \frac{3}{2})}{3} = 0$