Pre-Algebra Examples

$\frac{1}{2} x^{2} - \frac{1}{14} x - \frac{13}{7} = 0$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{2}}{2} - \frac{1}{14} x - \frac{13}{7} = 0$

Step 1.2

Combine $x$ and $\frac{1}{14}$ .

$\frac{x^{2}}{2} - \frac{x}{14} - \frac{13}{7} = 0$

Step 2

Multiply each term in $\frac{x^{2}}{2} - \frac{x}{14} - \frac{13}{7} = 0$ by $14$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x^{2}}{2} - \frac{x}{14} - \frac{13}{7} = 0$ by $14$ .

$\frac{x^{2}}{2} \cdot 14 - \frac{x}{14} \cdot 14 - \frac{13}{7} \cdot 14 = 0 \cdot 14$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Factor $2$ out of $14$ .

$\frac{x^{2}}{2} \cdot (2 (7)) - \frac{x}{14} \cdot 14 - \frac{13}{7} \cdot 14 = 0 \cdot 14$

Step 2.2.1.1.2

Cancel the common factor.

$\frac{x^{2}}{2} \cdot (2 \cdot 7) - \frac{x}{14} \cdot 14 - \frac{13}{7} \cdot 14 = 0 \cdot 14$

Step 2.2.1.1.3

Rewrite the expression.

$x^{2} \cdot 7 - \frac{x}{14} \cdot 14 - \frac{13}{7} \cdot 14 = 0 \cdot 14$

Step 2.2.1.2

Move $7$ to the left of $x^{2}$ .

$7 \cdot x^{2} - \frac{x}{14} \cdot 14 - \frac{13}{7} \cdot 14 = 0 \cdot 14$

Step 2.2.1.3

Cancel the common factor of $14$ .

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Step 2.2.1.3.1

Move the leading negative in $- \frac{x}{14}$ into the numerator.

$7 x^{2} + \frac{- x}{14} \cdot 14 - \frac{13}{7} \cdot 14 = 0 \cdot 14$

Step 2.2.1.3.2

Cancel the common factor.

$7 x^{2} + \frac{- x}{14} \cdot 14 - \frac{13}{7} \cdot 14 = 0 \cdot 14$

Step 2.2.1.3.3

Rewrite the expression.

$7 x^{2} - x - \frac{13}{7} \cdot 14 = 0 \cdot 14$

Step 2.2.1.4

Cancel the common factor of $7$ .

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Step 2.2.1.4.1

Move the leading negative in $- \frac{13}{7}$ into the numerator.

$7 x^{2} - x + \frac{- 13}{7} \cdot 14 = 0 \cdot 14$

Step 2.2.1.4.2

Factor $7$ out of $14$ .

$7 x^{2} - x + \frac{- 13}{7} \cdot (7 (2)) = 0 \cdot 14$

Step 2.2.1.4.3

Cancel the common factor.

$7 x^{2} - x + \frac{- 13}{7} \cdot (7 \cdot 2) = 0 \cdot 14$

Step 2.2.1.4.4

Rewrite the expression.

$7 x^{2} - x - 13 \cdot 2 = 0 \cdot 14$

Step 2.2.1.5

Multiply $- 13$ by $2$ .

$7 x^{2} - x - 26 = 0 \cdot 14$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $14$ .

$7 x^{2} - x - 26 = 0$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 7 \cdot - 26 = - 182$ and whose sum is $b = - 1$ .

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Step 3.1.1

Factor $- 1$ out of $- x$ .

$7 x^{2} - (x) - 26 = 0$

Step 3.1.2

Rewrite $- 1$ as $13$ plus $- 14$

$7 x^{2} + (13 - 14) x - 26 = 0$

Step 3.1.3

Apply the distributive property.

$7 x^{2} + 13 x - 14 x - 26 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(7 x^{2} + 13 x) - 14 x - 26 = 0$

Step 3.2.2

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

$x (7 x + 13) - 2 (7 x + 13) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $7 x + 13$ .

$(7 x + 13) (x - 2) = 0$

Step 4

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

$7 x + 13 = 0$

$x - 2 = 0$

Step 5

Set $7 x + 13$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $7 x + 13$ equal to $0$ .

$7 x + 13 = 0$

Step 5.2

Solve $7 x + 13 = 0$ for $x$ .

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Step 5.2.1

Subtract $13$ from both sides of the equation.

$7 x = - 13$

Step 5.2.2

Divide each term in $7 x = - 13$ by $7$ and simplify.

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Step 5.2.2.1

Divide each term in $7 x = - 13$ by $7$ .

$\frac{7 x}{7} = \frac{- 13}{7}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $7$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 13}{7}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 13}{7}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{13}{7}$

Step 6

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

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Step 6.1

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

$x - 2 = 0$

Step 6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 7

The final solution is all the values that make $(7 x + 13) (x - 2) = 0$ true.

$x = - \frac{13}{7}, 2$