Algebra Examples

$2 x^{- 2} - 19 x^{- 1} + 42 = 0$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 (\frac{1}{x^{2}}) - 19 x^{- 1} + 42 = 0$

Step 2

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

$\frac{2}{x^{2}} - 19 x^{- 1} + 42 = 0$

Step 3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{x^{2}} - 19 \frac{1}{x} + 42 = 0$

Step 4

Combine $- 19$ and $\frac{1}{x}$ .

$\frac{2}{x^{2}} + \frac{- 19}{x} + 42 = 0$

Step 5

Move the negative in front of the fraction.

$\frac{2}{x^{2}} - \frac{19}{x} + 42 = 0$

Step 6

Reorder terms.

$- \frac{19}{x} + \frac{2}{x^{2}} + 42 = 0$

Step 7

To write $- \frac{19}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{19}{x} \cdot \frac{x}{x} + \frac{2}{x^{2}} + 42 = 0$

Step 8

Write each expression with a common denominator of $x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{19}{x}$ by $\frac{x}{x}$ .

$- \frac{19 x}{x \cdot x} + \frac{2}{x^{2}} + 42 = 0$

Step 8.2

Raise $x$ to the power of $1$ .

$- \frac{19 x}{x \cdot x} + \frac{2}{x^{2}} + 42 = 0$

Step 8.3

Raise $x$ to the power of $1$ .

$- \frac{19 x}{x \cdot x} + \frac{2}{x^{2}} + 42 = 0$

Step 8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{19 x}{x^{1 + 1}} + \frac{2}{x^{2}} + 42 = 0$

Step 8.5

Add $1$ and $1$ .

$- \frac{19 x}{x^{2}} + \frac{2}{x^{2}} + 42 = 0$

Step 9

Combine the numerators over the common denominator.

$\frac{- 19 x + 2}{x^{2}} + 42 = 0$

Step 10

To write $42$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{- 19 x + 2}{x^{2}} + \frac{42 x^{2}}{x^{2}} = 0$

Step 11

Combine the numerators over the common denominator.

$\frac{- 19 x + 2 + 42 x^{2}}{x^{2}} = 0$

Step 12

Factor by grouping.

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Step 12.1

Reorder terms.

$\frac{42 x^{2} - 19 x + 2}{x^{2}} = 0$

Step 12.2

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 $a \cdot c = 42 \cdot 2 = 84$ and whose sum is $b = - 19$ .

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Step 12.2.1

Factor $- 19$ out of $- 19 x$ .

$\frac{42 x^{2} - 19 x + 2}{x^{2}} = 0$

Step 12.2.2

Rewrite $- 19$ as $- 7$ plus $- 12$

$\frac{42 x^{2} + (- 7 - 12) x + 2}{x^{2}} = 0$

Step 12.2.3

Apply the distributive property.

$\frac{42 x^{2} - 7 x - 12 x + 2}{x^{2}} = 0$

Step 12.3

Factor out the greatest common factor from each group.

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Step 12.3.1

Group the first two terms and the last two terms.

$\frac{(42 x^{2} - 7 x) - 12 x + 2}{x^{2}} = 0$

Step 12.3.2

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

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

Step 12.4

Factor the polynomial by factoring out the greatest common factor, $6 x - 1$ .

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

Step 13

Set the numerator equal to zero.

$(6 x - 1) (7 x - 2) = 0$

Step 14

Solve the equation for $x$ .

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Step 14.1

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

$6 x - 1 = 0$

$7 x - 2 = 0$

Step 14.2

Set $6 x - 1$ equal to $0$ and solve for $x$ .

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Step 14.2.1

Set $6 x - 1$ equal to $0$ .

$6 x - 1 = 0$

Step 14.2.2

Solve $6 x - 1 = 0$ for $x$ .

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Step 14.2.2.1

Add $1$ to both sides of the equation.

$6 x = 1$

Step 14.2.2.2

Divide each term in $6 x = 1$ by $6$ and simplify.

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Step 14.2.2.2.1

Divide each term in $6 x = 1$ by $6$ .

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

Step 14.2.2.2.2

Simplify the left side.

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Step 14.2.2.2.2.1

Cancel the common factor of $6$ .

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Step 14.2.2.2.2.1.1

Cancel the common factor.

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

Step 14.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 14.3

Set $7 x - 2$ equal to $0$ and solve for $x$ .

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Step 14.3.1

Set $7 x - 2$ equal to $0$ .

$7 x - 2 = 0$

Step 14.3.2

Solve $7 x - 2 = 0$ for $x$ .

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Step 14.3.2.1

Add $2$ to both sides of the equation.

$7 x = 2$

Step 14.3.2.2

Divide each term in $7 x = 2$ by $7$ and simplify.

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Step 14.3.2.2.1

Divide each term in $7 x = 2$ by $7$ .

$\frac{7 x}{7} = \frac{2}{7}$

Step 14.3.2.2.2

Simplify the left side.

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Step 14.3.2.2.2.1

Cancel the common factor of $7$ .

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Step 14.3.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{2}{7}$

Step 14.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{7}$

Step 14.4

The final solution is all the values that make $(6 x - 1) (7 x - 2) = 0$ true.

$x = \frac{1}{6}, \frac{2}{7}$