Calculus Examples

$\frac{x}{2} = 3 - \frac{17}{x}$

Step 1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 1.1

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

$2, 1, x$

Step 1.2

Since $2, 1, x$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 1, 1$ then find LCM for the variable part $x^{1}$ .

Step 1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.4

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 1.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.6

The LCM of $2, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 1.7

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 1.8

The LCM of $x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x$

Step 1.9

The LCM for $2, 1, x$ is the numeric part $2$ multiplied by the variable part.

$2 x$

Step 2

Multiply each term in $\frac{x}{2} = 3 - \frac{17}{x}$ by $2 x$ to eliminate the fractions.

Tap for more steps...

Step 2.1

Multiply each term in $\frac{x}{2} = 3 - \frac{17}{x}$ by $2 x$ .

$\frac{x}{2} (2 x) = 3 (2 x) - \frac{17}{x} (2 x)$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Rewrite using the commutative property of multiplication.

$2 \frac{x}{2} x = 3 (2 x) - \frac{17}{x} (2 x)$

Step 2.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.1

Cancel the common factor.

$2 \frac{x}{2} x = 3 (2 x) - \frac{17}{x} (2 x)$

Step 2.2.2.2

Rewrite the expression.

$x \cdot x = 3 (2 x) - \frac{17}{x} (2 x)$

Step 2.2.3

Multiply $x$ by $x$ .

$x^{2} = 3 (2 x) - \frac{17}{x} (2 x)$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Multiply $2$ by $3$ .

$x^{2} = 6 x - \frac{17}{x} (2 x)$

Step 2.3.1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.3.1.2.1

Move the leading negative in $- \frac{17}{x}$ into the numerator.

$x^{2} = 6 x + \frac{- 17}{x} (2 x)$

Step 2.3.1.2.2

Factor $x$ out of $2 x$ .

$x^{2} = 6 x + \frac{- 17}{x} (x \cdot 2)$

Step 2.3.1.2.3

Cancel the common factor.

$x^{2} = 6 x + \frac{- 17}{x} (x \cdot 2)$

Step 2.3.1.2.4

Rewrite the expression.

$x^{2} = 6 x - 17 \cdot 2$

Step 2.3.1.3

Multiply $- 17$ by $2$ .

$x^{2} = 6 x - 34$

Step 3

Solve the equation.

Tap for more steps...

Step 3.1

Subtract $6 x$ from both sides of the equation.

$x^{2} - 6 x = - 34$

Step 3.2

Add $34$ to both sides of the equation.

$x^{2} - 6 x + 34 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

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

Step 3.4

Substitute the values $a = 1$ , $b = - 6$ , and $c = 34$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 34)}}{2 \cdot 1}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Simplify the numerator.

Tap for more steps...

Step 3.5.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 34}}{2 \cdot 1}$

Step 3.5.1.2

Multiply $- 4 \cdot 1 \cdot 34$ .

Tap for more steps...

Step 3.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 34}}{2 \cdot 1}$

Step 3.5.1.2.2

Multiply $- 4$ by $34$ .

$x = \frac{6 \pm \sqrt{36 - 136}}{2 \cdot 1}$

Step 3.5.1.3

Subtract $136$ from $36$ .

$x = \frac{6 \pm \sqrt{- 100}}{2 \cdot 1}$

Step 3.5.1.4

Rewrite $- 100$ as $- 1 (100)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

Step 3.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{6 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Step 3.5.1.7

Rewrite $100$ as $10^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

Step 3.5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{6 \pm i \cdot 10}{2 \cdot 1}$

Step 3.5.1.9

Move $10$ to the left of $i$ .

$x = \frac{6 \pm 10 i}{2 \cdot 1}$

Step 3.5.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 10 i}{2}$

Step 3.5.3

Simplify $\frac{6 \pm 10 i}{2}$ .

$x = 3 \pm 5 i$

Step 3.6

The final answer is the combination of both solutions.

$x = 3 + 5 i, 3 - 5 i$

$x = 3 \pm 5 i$

Step 4