Solve for x 1/x+2x=3

$\frac{1}{x} + 2 x = 3$

Step 1

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.

$x$

Step 2

Multiply each term in $\frac{1}{x} + 2 x = 3$ by $x$ to eliminate the fractions.

Multiply each term in $\frac{1}{x} + 2 x = 3$ by $x$ .

$\frac{1}{x} x + 2 x \cdot x = 3 x$

Simplify the left side.

Simplify each term.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{1}{x} x + 2 x \cdot x = 3 x$

Rewrite the expression.

$1 + 2 x \cdot x = 3 x$

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

Move $x$ .

$1 + 2 (x \cdot x) = 3 x$

Multiply $x$ by $x$ .

$1 + 2 x^{2} = 3 x$

Step 3

Solve the equation.

Subtract $3 x$ from both sides of the equation.

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

Factor by grouping.

Reorder terms.

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

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 $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = - 3$ .

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

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

Rewrite $- 3$ as $- 1$ plus $- 2$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

$2 x - 1 = 0$

$x - 1 = 0$

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

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

$2 x - 1 = 0$

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

Add $1$ to both sides of the equation.

$2 x = 1$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5, 1$

Solve for x 1/x+2x=3

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