Solve the Rational Equation for x 4/x+1/(x^2)=1/(5x^2)

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

Since ${\displaystyle x,x^2,5x^2}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,1,5}$ then find LCM for the variable part ${\displaystyle x^1,x^2,x^2}$.

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.

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

Not prime

Since ${\displaystyle 5}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 5}$.

${\displaystyle 5}$ is a prime number

The LCM of ${\displaystyle 1,1,5}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

${\displaystyle x}$ occurs ${\displaystyle 1}$ time.

The factors for ${\displaystyle x^2}$ are ${\displaystyle x\cdot x}$, which is ${\displaystyle x}$ multiplied by each other ${\displaystyle 2}$ times.

${\displaystyle x}$ occurs ${\displaystyle 2}$ times.

The LCM of ${\displaystyle x^1,x^2,x^2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

Multiply ${\displaystyle x}$ by ${\displaystyle x}$.

The LCM for ${\displaystyle x,x^2,5x^2}$ is the numeric part ${\displaystyle 5}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{4}{x}+\frac{1}{x^2}=\frac{1}{5x^2}}$ by ${\displaystyle 5x^2}$.

Simplify the left side.

Simplify the right side.

Move all terms not containing ${\displaystyle x}$ to the right side of the equation.

Divide each term in ${\displaystyle 20x=-4}$ by ${\displaystyle 20}$ and simplify.