On a second report that you reviewed, the team simplified the expression to 4. Show how to simplify the rational expression correctly. Explain your steps so that the team understands how to simplify an expression similar to this in future experiments.
To simplify a rational expression, we need to simplify both the numerator and the denominator separately and then combine the simplified terms.
Let's assume the given expression is:
(x^2 - 9x + 20) / (x^2 - 5x - 24)
To simplify the numerator, we need to find the factors of the quadratic equation x^2 - 9x + 20, which can be factored as (x - 4)(x - 5).
So, the numerator becomes (x - 4)(x - 5)
Similarly, let's factorize the denominator, x^2 - 5x - 24, which can be factored as (x - 8)(x + 3).
So, the denominator becomes (x - 8)(x + 3).
Now, the simplified expression is:
[(x - 4)(x - 5)] / [(x - 8)(x + 3)]
But the goal is to simplify the expression to 4.
To get the given expression to equal 4, we need to equate the numerator to 4 and the denominator to 1.
Setting the numerator equal to 4, we have:
(x - 4)(x - 5) = 4
Expanding the equation, we get:
x^2 - 9x + 20 = 4
Rearranging the terms, we have:
x^2 - 9x + 16 = 0
Now, we find the factors of this quadratic equation, which are (x - 8)(x - 2).
Setting the denominator equal to 1, we have:
(x - 8)(x + 3) = 1
Again, expanding the equation, we get:
x^2 - 5x - 24 = 1
Rearranging the terms, we have:
x^2 - 5x - 25 = 0
Now, we find the factors of this quadratic equation, which are (x - 5)(x + 5).
So, the simplified expression is:
[(x - 4)(x - 5)] / [(x - 8)(x + 3)] = 4 / 1
Therefore, the correct simplification of the given expression is:
(x - 4)(x - 5) = 4(x - 8)(x + 3)
By following these steps, the team can simplify similar rational expressions in future experiments.