Pre-Algebra Examples

$x^{2} - \frac{7}{6} x + \frac{1}{3} = 0$

Step 1

Simplify each term.

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Step 1.1

Combine $x$ and $\frac{7}{6}$ .

$x^{2} - \frac{x \cdot 7}{6} + \frac{1}{3} = 0$

Step 1.2

Move $7$ to the left of $x$ .

$x^{2} - \frac{7 x}{6} + \frac{1}{3} = 0$

Step 2

Multiply through by the least common denominator $6$ , then simplify.

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Step 2.1

Apply the distributive property.

$6 x^{2} + 6 (- \frac{7 x}{6}) + 6 (\frac{1}{3}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{7 x}{6}$ into the numerator.

$6 x^{2} + 6 (\frac{- 7 x}{6}) + 6 (\frac{1}{3}) = 0$

Step 2.2.1.2

Cancel the common factor.

$6 x^{2} + 6 (\frac{- 7 x}{6}) + 6 (\frac{1}{3}) = 0$

Step 2.2.1.3

Rewrite the expression.

$6 x^{2} - 7 x + 6 (\frac{1}{3}) = 0$

Step 2.2.2

Cancel the common factor of $3$ .

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Step 2.2.2.1

Factor $3$ out of $6$ .

$6 x^{2} - 7 x + 3 (2) (\frac{1}{3}) = 0$

Step 2.2.2.2

Cancel the common factor.

$6 x^{2} - 7 x + 3 \cdot (2 (\frac{1}{3})) = 0$

Step 2.2.2.3

Rewrite the expression.

$6 x^{2} - 7 x + 2 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 6$ , $b = - 7$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (6 \cdot 2)}}{2 \cdot 6}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$

Step 5.1.2

Multiply $- 4 \cdot 6 \cdot 2$ .

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Step 5.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{7 \pm \sqrt{49 - 24 \cdot 2}}{2 \cdot 6}$

Step 5.1.2.2

Multiply $- 24$ by $2$ .

$x = \frac{7 \pm \sqrt{49 - 48}}{2 \cdot 6}$

Step 5.1.3

Subtract $48$ from $49$ .

$x = \frac{7 \pm \sqrt{1}}{2 \cdot 6}$

Step 5.1.4

Any root of $1$ is $1$ .

$x = \frac{7 \pm 1}{2 \cdot 6}$

Step 5.2

Multiply $2$ by $6$ .

$x = \frac{7 \pm 1}{12}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$

Step 6.1.2

Multiply $- 4 \cdot 6 \cdot 2$ .

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Step 6.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{7 \pm \sqrt{49 - 24 \cdot 2}}{2 \cdot 6}$

Step 6.1.2.2

Multiply $- 24$ by $2$ .

$x = \frac{7 \pm \sqrt{49 - 48}}{2 \cdot 6}$

Step 6.1.3

Subtract $48$ from $49$ .

$x = \frac{7 \pm \sqrt{1}}{2 \cdot 6}$

Step 6.1.4

Any root of $1$ is $1$ .

$x = \frac{7 \pm 1}{2 \cdot 6}$

Step 6.2

Multiply $2$ by $6$ .

$x = \frac{7 \pm 1}{12}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{7 + 1}{12}$

Step 6.4

Add $7$ and $1$ .

$x = \frac{8}{12}$

Step 6.5

Cancel the common factor of $8$ and $12$ .

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Step 6.5.1

Factor $4$ out of $8$ .

$x = \frac{4 (2)}{12}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Factor $4$ out of $12$ .

$x = \frac{4 \cdot 2}{4 \cdot 3}$

Step 6.5.2.2

Cancel the common factor.

$x = \frac{4 \cdot 2}{4 \cdot 3}$

Step 6.5.2.3

Rewrite the expression.

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

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$

Step 7.1.2

Multiply $- 4 \cdot 6 \cdot 2$ .

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Step 7.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{7 \pm \sqrt{49 - 24 \cdot 2}}{2 \cdot 6}$

Step 7.1.2.2

Multiply $- 24$ by $2$ .

$x = \frac{7 \pm \sqrt{49 - 48}}{2 \cdot 6}$

Step 7.1.3

Subtract $48$ from $49$ .

$x = \frac{7 \pm \sqrt{1}}{2 \cdot 6}$

Step 7.1.4

Any root of $1$ is $1$ .

$x = \frac{7 \pm 1}{2 \cdot 6}$

Step 7.2

Multiply $2$ by $6$ .

$x = \frac{7 \pm 1}{12}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{7 - 1}{12}$

Step 7.4

Subtract $1$ from $7$ .

$x = \frac{6}{12}$

Step 7.5

Cancel the common factor of $6$ and $12$ .

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Step 7.5.1

Factor $6$ out of $6$ .

$x = \frac{6 (1)}{12}$

Step 7.5.2

Cancel the common factors.

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Step 7.5.2.1

Factor $6$ out of $12$ .

$x = \frac{6 \cdot 1}{6 \cdot 2}$

Step 7.5.2.2

Cancel the common factor.

$x = \frac{6 \cdot 1}{6 \cdot 2}$

Step 7.5.2.3

Rewrite the expression.

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

Step 8

The final answer is the combination of both solutions.

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