Solve by Factoring 15x-16=15/x

$15 x - 16 = \frac{15}{x}$

Step 1

Subtract $\frac{15}{x}$ from both sides of the equation.

$15 x - 16 - \frac{15}{x} = 0$

Step 2

Reorder terms.

$15 x - \frac{15}{x} - 16 = 0$

Step 3

To write $15 x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{15 x \cdot x}{x} - \frac{15}{x} - 16 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{15 x \cdot x - 15}{x} - 16 = 0$

Step 5

Simplify the numerator.

Factor $15$ out of $15 x \cdot x - 15$ .

Factor $15$ out of $15 x \cdot x$ .

$\frac{15 (x \cdot x) - 15}{x} - 16 = 0$

Factor $15$ out of $- 15$ .

$\frac{15 (x \cdot x) + 15 (- 1)}{x} - 16 = 0$

Factor $15$ out of $15 (x \cdot x) + 15 (- 1)$ .

$\frac{15 (x \cdot x - 1)}{x} - 16 = 0$

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

$\frac{15 (x \cdot x - 1)}{x} - 16 = 0$

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

$\frac{15 (x \cdot x - 1)}{x} - 16 = 0$

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

$\frac{15 (x^{1 + 1} - 1)}{x} - 16 = 0$

Add $1$ and $1$ .

$\frac{15 (x^{2} - 1)}{x} - 16 = 0$

Rewrite $1$ as $1^{2}$ .

$\frac{15 (x^{2} - 1^{2})}{x} - 16 = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{15 ((x + 1) (x - 1))}{x} - 16 = 0$

Remove unnecessary parentheses.

$\frac{15 (x + 1) (x - 1)}{x} - 16 = 0$

Step 6

To write $- 16$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{15 (x + 1) (x - 1)}{x} - 16 \cdot \frac{x}{x} = 0$

Step 7

Combine $- 16$ and $\frac{x}{x}$ .

$\frac{15 (x + 1) (x - 1)}{x} + \frac{- 16 x}{x} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{15 (x + 1) (x - 1) - 16 x}{x} = 0$

Step 9

Simplify the numerator.

Apply the distributive property.

$\frac{(15 x + 15 \cdot 1) (x - 1) - 16 x}{x} = 0$

Multiply $15$ by $1$ .

$\frac{(15 x + 15) (x - 1) - 16 x}{x} = 0$

Expand $(15 x + 15) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{15 x (x - 1) + 15 (x - 1) - 16 x}{x} = 0$

Apply the distributive property.

$\frac{15 x \cdot x + 15 x \cdot - 1 + 15 (x - 1) - 16 x}{x} = 0$

Apply the distributive property.

$\frac{15 x \cdot x + 15 x \cdot - 1 + 15 x + 15 \cdot - 1 - 16 x}{x} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{15 (x \cdot x) + 15 x \cdot - 1 + 15 x + 15 \cdot - 1 - 16 x}{x} = 0$

Multiply $x$ by $x$ .

$\frac{15 x^{2} + 15 x \cdot - 1 + 15 x + 15 \cdot - 1 - 16 x}{x} = 0$

Multiply $- 1$ by $15$ .

$\frac{15 x^{2} - 15 x + 15 x + 15 \cdot - 1 - 16 x}{x} = 0$

Multiply $15$ by $- 1$ .

$\frac{15 x^{2} - 15 x + 15 x - 15 - 16 x}{x} = 0$

Add $- 15 x$ and $15 x$ .

$\frac{15 x^{2} + 0 - 15 - 16 x}{x} = 0$

Add $15 x^{2}$ and $0$ .

$\frac{15 x^{2} - 15 - 16 x}{x} = 0$

Reorder terms.

$\frac{15 x^{2} - 16 x - 15}{x} = 0$

Factor by grouping.

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 $a \cdot c = 15 \cdot - 15 = - 225$ and whose sum is $b = - 16$ .

Factor $- 16$ out of $- 16 x$ .

$\frac{15 x^{2} - 16 x - 15}{x} = 0$

Rewrite $- 16$ as $9$ plus $- 25$

$\frac{15 x^{2} + (9 - 25) x - 15}{x} = 0$

Apply the distributive property.

$\frac{15 x^{2} + 9 x - 25 x - 15}{x} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(15 x^{2} + 9 x) - 25 x - 15}{x} = 0$

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

$\frac{3 x (5 x + 3) - 5 (5 x + 3)}{x} = 0$

Factor the polynomial by factoring out the greatest common factor, $5 x + 3$ .

$\frac{(5 x + 3) (3 x - 5)}{x} = 0$

Step 10

Set the numerator equal to zero.

$(5 x + 3) (3 x - 5) = 0$

Step 11

Solve the equation for $x$ .

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

$5 x + 3 = 0$

$3 x - 5 = 0$

Set $5 x + 3$ equal to $0$ and solve for $x$ .

Set $5 x + 3$ equal to $0$ .

$5 x + 3 = 0$

Solve $5 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$5 x = - 3$

Divide each term in $5 x = - 3$ by $5$ and simplify.

Divide each term in $5 x = - 3$ by $5$ .

$\frac{5 x}{5} = \frac{- 3}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 3}{5}$

Divide $x$ by $1$ .

$x = \frac{- 3}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{5}$

Set $3 x - 5$ equal to $0$ and solve for $x$ .

Set $3 x - 5$ equal to $0$ .

$3 x - 5 = 0$

Solve $3 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$3 x = 5$

Divide each term in $3 x = 5$ by $3$ and simplify.

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Divide $x$ by $1$ .

$x = \frac{5}{3}$

The final solution is all the values that make $(5 x + 3) (3 x - 5) = 0$ true.

$x = - \frac{3}{5}, \frac{5}{3}$

Solve by Factoring 15x-16=15/x

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