Question

Which expression is equivalent to (see picture) if no denominator equals zero?

Answer 1

Explanation:

The expression that we have is:

`((x-1)^2)/(x^2-x-12)\cdot(x^2+x-6)/(x^2-6x+5)`

And we need to find the equivalent expression.

Step 1. First, we expand the term (x-1)^2 as follows:

`((x-1)(x-1))/(x^2-x-12)\cdot(x^(2)+x-6)/(x^(2)-6x+5)`

Then, we need to factor the rest of the three quadratic expressions.

Let's review the general process:

-For quadratic expressions of the form:

`x^2+bx+c`

We factor it by finding two numbers that when you add them the result is b and when you multiply them the result is c.

Step 2. Factoring the three quadratic expressions:

`x^2-x-12=(x-4)(x+3)`
`x^2+x-6=(x-2)(x+3)`
`x^2-6x+5=(x-5)(x-1)`

Step 3. Using the factored expressions, the result is:

We cancel the terms that are both in the numerator and in the denominator:

And we are left only with the following terms:

`((x-1)(x-2))/((x-4)(x-5))`

Step 4. Multiplying the terms to find the final expression:

`((x-1)(x-2))/((x-4)(x-5))=(x^2-2x-x+2)/(x^2-5x-4x+20)`

Combining like terms:

`(x^2-3x+2)/(x^2-9x+20)`

This is shown in option A.

Answer:

`(x^(2)-3x+2)/(x^(2)-9x+20)`