# ACT Math : How to find the solution of a rational equation with a binomial denominator

## Example Questions

### Example Question #1 : How To Find The Solution Of A Rational Equation With A Binomial Denominator

For the equation , what is(are) the solution(s) for ?

Explanation:

, can be factored to (x -7)(x-3) = 0. Therefore, x-7 = 0 and x-3 = 0. Solving for x in both cases, gives 7 and 3.

### Example Question #2 : How To Find The Solution Of A Rational Equation With A Binomial Denominator

Simplify:

Explanation:

In order to begin this kind of a problem, it's key to look at parts of the rational expression that can be simplified.

In this case, the denominator is an already-simplified binomial; however, the numerator can be factored.

The roots will be numbers that sum up to  but have the product of .

The options include:

When these options are summed up:

We can negate the last three options because the first option of  and  fulfill the requirements. Therefore, the numerator can be factored into the following:

Because the quantity  appears in the denominator, this can be "canceled out." This leaves the final answer to be the quantity .

### Example Question #3 : How To Find The Solution Of A Rational Equation With A Binomial Denominator

Simplify:

Explanation:

In order to begin this kind of a problem, it's key to look at parts of the rational expression that can be simplified.

In this case, the denominator is an already-simplified binomial; however, the numerator can be factored through "factoring by grouping." This can be a helpful idea to keep in mind when you come across a polynomial with four terms and simplifying is involved.

can be simplified first by removing the common factor of  from the first two terms and the common factor of  from the last two terms:

This leaves two terms that are identical  and their coefficients, which can be combined into another term to complete the factoring:

Consider the denominator; the quantity  appears, so the  in the numerator and in the denominator can be cancelled out. The simplified expression is then left as .