Solve for x 3-1/x=-4x

$3 - \frac{1}{x} = - 4 x$

Subtract $3$ from both sides of the equation.

$- \frac{1}{x} = - 4 x - 3$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1, 1$

The LCM of one and any expression is the expression.

$x$

Multiply each term in $- \frac{1}{x} = - 4 x - 3$ by $x$ to eliminate the fractions.

Multiply each term in $- \frac{1}{x} = - 4 x - 3$ by $x$ .

$- \frac{1}{x} x = - 4 x \cdot x - 3 x$

Simplify the left side.

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{- 1}{x} x = - 4 x \cdot x - 3 x$

Cancel the common factor.

$\frac{- 1}{x} x = - 4 x \cdot x - 3 x$

Rewrite the expression.

$- 1 = - 4 x \cdot x - 3 x$

Simplify the right side.

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

Move $x$ .

$- 1 = - 4 (x \cdot x) - 3 x$

Multiply $x$ by $x$ .

$- 1 = - 4 x^{2} - 3 x$

Solve the equation.

Rewrite the equation as $- 4 x^{2} - 3 x = - 1$ .

$- 4 x^{2} - 3 x = - 1$

Add $1$ to both sides of the equation.

$- 4 x^{2} - 3 x + 1 = 0$

Factor the left side of the equation.

Factor $- 1$ out of $- 4 x^{2} - 3 x + 1$ .

Factor $- 1$ out of $- 4 x^{2}$ .

$- (4 x^{2}) - 3 x + 1 = 0$

Factor $- 1$ out of $- 3 x$ .

$- (4 x^{2}) - (3 x) + 1 = 0$

Rewrite $1$ as $- 1 (- 1)$ .

$- (4 x^{2}) - (3 x) - 1 \cdot - 1 = 0$

Factor $- 1$ out of $- (4 x^{2}) - (3 x)$ .

$- (4 x^{2} + 3 x) - 1 \cdot - 1 = 0$

Factor $- 1$ out of $- (4 x^{2} + 3 x) - 1 (- 1)$ .

$- (4 x^{2} + 3 x - 1) = 0$

Factor.

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 $a \cdot c = 4 \cdot - 1 = - 4$ and whose sum is $b = 3$ .

Factor $3$ out of $3 x$ .

$- (4 x^{2} + 3 (x) - 1) = 0$

Rewrite $3$ as $- 1$ plus $4$

$- (4 x^{2} + (- 1 + 4) x - 1) = 0$

Apply the distributive property.

$- (4 x^{2} - 1 x + 4 x - 1) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- ((4 x^{2} - 1 x) + 4 x - 1) = 0$

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

$- (x (4 x - 1) + 1 (4 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$- ((4 x - 1) (x + 1)) = 0$

Remove unnecessary parentheses.

$- (4 x - 1) (x + 1) = 0$

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

$4 x - 1 = 0$

$x + 1 = 0$

Set $4 x - 1$ equal to $0$ and solve for $x$ .

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Solve $4 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $- (4 x - 1) (x + 1) = 0$ true.

$x = \frac{1}{4}, - 1$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}, - 1$

Decimal Form:

$x = 0.25, - 1$

Solve for x 3-1/x=-4x

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