Algebra Examples

$\frac{4}{x^{2}} - \frac{36}{x} = 19$

Step 1

Subtract $19$ from both sides of the equation.

$\frac{4}{x^{2}} - \frac{36}{x} - 19 = 0$

Step 2

Reorder terms.

$- \frac{36}{x} + \frac{4}{x^{2}} - 19 = 0$

Step 3

To write $- \frac{36}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{36}{x} \cdot \frac{x}{x} + \frac{4}{x^{2}} - 19 = 0$

Step 4

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

Tap for more steps...

Step 4.1

Multiply $\frac{36}{x}$ by $\frac{x}{x}$ .

$- \frac{36 x}{x \cdot x} + \frac{4}{x^{2}} - 19 = 0$

Step 4.2

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

$- \frac{36 x}{x \cdot x} + \frac{4}{x^{2}} - 19 = 0$

Step 4.3

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

$- \frac{36 x}{x \cdot x} + \frac{4}{x^{2}} - 19 = 0$

Step 4.4

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

$- \frac{36 x}{x^{1 + 1}} + \frac{4}{x^{2}} - 19 = 0$

Step 4.5

Add $1$ and $1$ .

$- \frac{36 x}{x^{2}} + \frac{4}{x^{2}} - 19 = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{- 36 x + 4}{x^{2}} - 19 = 0$

Step 6

Factor $4$ out of $- 36 x + 4$ .

Tap for more steps...

Step 6.1

Factor $4$ out of $- 36 x$ .

$\frac{4 (- 9 x) + 4}{x^{2}} - 19 = 0$

Step 6.2

Factor $4$ out of $4$ .

$\frac{4 (- 9 x) + 4 (1)}{x^{2}} - 19 = 0$

Step 6.3

Factor $4$ out of $4 (- 9 x) + 4 (1)$ .

$\frac{4 (- 9 x + 1)}{x^{2}} - 19 = 0$

Step 7

To write $- 19$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{4 (- 9 x + 1)}{x^{2}} - 19 \cdot \frac{x^{2}}{x^{2}} = 0$

Step 8

Combine $- 19$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{4 (- 9 x + 1)}{x^{2}} + \frac{- 19 x^{2}}{x^{2}} = 0$

Step 9

Combine the numerators over the common denominator.

$\frac{4 (- 9 x + 1) - 19 x^{2}}{x^{2}} = 0$

Step 10

Simplify the numerator.

Tap for more steps...

Step 10.1

Apply the distributive property.

$\frac{4 (- 9 x) + 4 \cdot 1 - 19 x^{2}}{x^{2}} = 0$

Step 10.2

Multiply $- 9$ by $4$ .

$\frac{- 36 x + 4 \cdot 1 - 19 x^{2}}{x^{2}} = 0$

Step 10.3

Multiply $4$ by $1$ .

$\frac{- 36 x + 4 - 19 x^{2}}{x^{2}} = 0$

Step 10.4

Reorder terms.

$\frac{- 19 x^{2} - 36 x + 4}{x^{2}} = 0$

Step 10.5

Factor by grouping.

Tap for more steps...

Step 10.5.1

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 $a \cdot c = - 19 \cdot 4 = - 76$ and whose sum is $b = - 36$ .

Tap for more steps...

Step 10.5.1.1

Factor $- 36$ out of $- 36 x$ .

$\frac{- 19 x^{2} - 36 x + 4}{x^{2}} = 0$

Step 10.5.1.2

Rewrite $- 36$ as $2$ plus $- 38$

$\frac{- 19 x^{2} + (2 - 38) x + 4}{x^{2}} = 0$

Step 10.5.1.3

Apply the distributive property.

$\frac{- 19 x^{2} + 2 x - 38 x + 4}{x^{2}} = 0$

Step 10.5.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 10.5.2.1

Group the first two terms and the last two terms.

$\frac{(- 19 x^{2} + 2 x) - 38 x + 4}{x^{2}} = 0$

Step 10.5.2.2

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

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

Step 10.5.3

Factor the polynomial by factoring out the greatest common factor, $- 19 x + 2$ .

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

Step 11

Set the numerator equal to zero.

$(- 19 x + 2) (x + 2) = 0$

Step 12

Solve the equation for $x$ .

Tap for more steps...

Step 12.1

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

$- 19 x + 2 = 0$

$x + 2 = 0$

Step 12.2

Set $- 19 x + 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 12.2.1

Set $- 19 x + 2$ equal to $0$ .

$- 19 x + 2 = 0$

Step 12.2.2

Solve $- 19 x + 2 = 0$ for $x$ .

Tap for more steps...

Step 12.2.2.1

Subtract $2$ from both sides of the equation.

$- 19 x = - 2$

Step 12.2.2.2

Divide each term in $- 19 x = - 2$ by $- 19$ and simplify.

Tap for more steps...

Step 12.2.2.2.1

Divide each term in $- 19 x = - 2$ by $- 19$ .

$\frac{- 19 x}{- 19} = \frac{- 2}{- 19}$

Step 12.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 12.2.2.2.2.1

Cancel the common factor of $- 19$ .

Tap for more steps...

Step 12.2.2.2.2.1.1

Cancel the common factor.

$\frac{- 19 x}{- 19} = \frac{- 2}{- 19}$

Step 12.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 19}$

Step 12.2.2.2.3

Simplify the right side.

Tap for more steps...

Step 12.2.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{2}{19}$

Step 12.3

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

Tap for more steps...

Step 12.3.1

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

$x + 2 = 0$

Step 12.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 12.4

The final solution is all the values that make $(- 19 x + 2) (x + 2) = 0$ true.

$x = \frac{2}{19}, - 2$