Algebra Examples

$\frac{3}{x} + \frac{4}{2 x} = x - 1$

Step 1

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 1.1

Simplify the left side.

Tap for more steps...

Step 1.1.1

Simplify $\frac{3}{x} + \frac{4}{2 x}$ .

Tap for more steps...

Step 1.1.1.1

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 1.1.1.1.1

Factor $2$ out of $4$ .

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

Step 1.1.1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 1.1.1.1.2.2

Cancel the common factor.

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

Step 1.1.1.1.2.3

Rewrite the expression.

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

Step 1.1.1.2

Combine the numerators over the common denominator.

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

Step 1.1.1.3

Add $3$ and $2$ .

$\frac{5}{x} = x - 1$

Step 1.2

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 1.2.1

Subtract $x$ from both sides of the equation.

$\frac{5}{x} - x = - 1$

Step 1.2.2

Add $1$ to both sides of the equation.

$\frac{5}{x} - x + 1 = 0$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

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

$x, 1, 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x$

Step 3

Multiply each term in $\frac{5}{x} - x + 1 = 0$ by $x$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{5}{x} - x + 1 = 0$ by $x$ .

$\frac{5}{x} x - x \cdot x + 1 x = 0 x$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.2.1.1.1

Cancel the common factor.

$\frac{5}{x} x - x \cdot x + 1 x = 0 x$

Step 3.2.1.1.2

Rewrite the expression.

$5 - x \cdot x + 1 x = 0 x$

Step 3.2.1.2

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

Tap for more steps...

Step 3.2.1.2.1

Move $x$ .

$5 - (x \cdot x) + 1 x = 0 x$

Step 3.2.1.2.2

Multiply $x$ by $x$ .

$5 - x^{2} + 1 x = 0 x$

Step 3.2.1.3

Multiply $x$ by $1$ .

$5 - x^{2} + x = 0 x$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply $0$ by $x$ .

$5 - x^{2} + x = 0$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.2

Substitute the values $a = - 1$ , $b = 1$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 1 \cdot 5)}}{2 \cdot - 1}$

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

Simplify the numerator.

Tap for more steps...

Step 4.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 5}}{2 \cdot - 1}$

Step 4.3.1.2

Multiply $- 4 \cdot - 1 \cdot 5$ .

Tap for more steps...

Step 4.3.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4 \cdot 5}}{2 \cdot - 1}$

Step 4.3.1.2.2

Multiply $4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 + 20}}{2 \cdot - 1}$

Step 4.3.1.3

Add $1$ and $20$ .

$x = \frac{- 1 \pm \sqrt{21}}{2 \cdot - 1}$

Step 4.3.2

Multiply $2$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{21}}{- 2}$

Step 4.3.3

Simplify $\frac{- 1 \pm \sqrt{21}}{- 2}$ .

$x = \frac{1 \pm \sqrt{21}}{2}$

Step 4.4

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{21}}{2}, \frac{1 - \sqrt{21}}{2}$

$x = \frac{1 \pm \sqrt{21}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 \pm \sqrt{21}}{2}$

Decimal Form:

$x = 2.79128784 \dots, - 1.79128784 \dots$