Algebra Examples

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

Step 1

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

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

Step 2

Combine $3$ and $\frac{1}{x^{2}}$ .

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

Step 3

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

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

Step 4

Combine $7$ and $\frac{1}{x}$ .

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

Step 5

Reorder terms.

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

Step 6

To write $\frac{7}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 7

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

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Step 7.1

Multiply $\frac{7}{x}$ by $\frac{x}{x}$ .

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

Step 7.2

Multiply $x$ by $x$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Factor by grouping.

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Step 11.1

Reorder terms.

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

Step 11.2

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = 7$ .

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Step 11.2.1

Factor $7$ out of $7 x$ .

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

Step 11.2.2

Rewrite $7$ as $1$ plus $6$

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

Step 11.2.3

Apply the distributive property.

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

Step 11.2.4

Multiply $x$ by $1$ .

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

Step 11.3

Factor out the greatest common factor from each group.

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Step 11.3.1

Group the first two terms and the last two terms.

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

Step 11.3.2

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

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

Step 11.4

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Step 12

Set the numerator equal to zero.

$(2 x + 1) (x + 3) = 0$

Step 13

Solve the equation for $x$ .

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Step 13.1

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

$2 x + 1 = 0$

$x + 3 = 0$

Step 13.2

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 13.2.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 13.2.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 13.2.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 13.2.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 13.2.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 13.2.2.2.2

Simplify the left side.

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Step 13.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 13.2.2.2.2.1.1

Cancel the common factor.

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

Step 13.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 13.2.2.2.3

Simplify the right side.

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Step 13.2.2.2.3.1

Move the negative in front of the fraction.

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

Step 13.3

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 13.3.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 13.3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 13.4

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

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