Solve for z (6z)/(z^2+4z-21)+7/(z+7)=1

Consider the form ${\displaystyle x^2+bx+c}$. Find a pair of integers whose product is ${\displaystyle c}$ and whose sum is ${\displaystyle b}$. In this case, whose product is ${\displaystyle -21}$ and whose sum is ${\displaystyle 4}$.

Write the factored form using these integers.

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

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

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

The factor for ${\displaystyle z-3}$ is ${\displaystyle z-3}$ itself.

${\displaystyle (z-3)}$ occurs ${\displaystyle 1}$ time.

The factor for ${\displaystyle z+7}$ is ${\displaystyle z+7}$ itself.

${\displaystyle (z+7)}$ occurs ${\displaystyle 1}$ time.

The LCM of ${\displaystyle z-3,z+7,z+7}$ is the result of multiplying all factors the greatest number of times they occur in either term.

Multiply each term in ${\displaystyle \frac{6z}{(z-3)(z+7)}+\frac{7}{z+7}=1}$ by ${\displaystyle (z-3)(z+7)}$ in order to remove all the denominators from the equation.

Simplify ${\displaystyle \frac{6z}{(z-3)(z+7)}\cdot \left((z-3)(z+7)\right)+\frac{7}{z+7}\cdot \left((z-3)(z+7)\right)}$.

Simplify ${\displaystyle 1\cdot \left((z-3)(z+7)\right)}$.

Since ${\displaystyle z}$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

Move all terms containing ${\displaystyle z}$ to the left side of the equation.

Move ${\displaystyle 21}$ to the left side of the equation by adding it to both sides.

Combine the opposite terms in ${\displaystyle z^2-9z-21+21}$.

Factor ${\displaystyle z}$ out of ${\displaystyle z^2-9z}$.

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

Set the first factor equal to ${\displaystyle 0}$.

Set the next factor equal to ${\displaystyle 0}$ and solve.

The final solution is all the values that make ${\displaystyle z(z-9)=0}$ true.