Simplify x^2 - 9x + 18/ x^2 - 6x + 9 for all values of x for which the expression is defined.
A. x-3/x-6
B. x-6/x-3
C. x-1/x+9
D. x+9/x-1
Please explain how you work this problem in detail. Thanks.
Assuming you mean
(x^2 - 9x + 18)/ (x^2 - 6x + 9)
= (x-6)(x-3)/((x-3)(x-3))
= (x-6)/(x-3) , x ≠ 3
which would have been B) with proper brackets.
To simplify the given expression, we can factor the numerator and denominator separately and then cancel out any common factors.
Let's start by factoring the numerator:
x^2 - 9x + 18
We are looking for two numbers that multiply to give 18 and add to give -9 (the coefficient of the x term). The numbers that satisfy these conditions are -3 and -6. Therefore, we can rewrite the numerator as:
(x - 3)(x - 6)
Now let's factor the denominator:
x^2 - 6x + 9
Again, we are looking for two numbers that multiply to give 9 and add to give -6. The numbers that satisfy these conditions are -3 and -3. Hence, we can factor the denominator as:
(x - 3)(x - 3)
Now we have the simplified expression as:
(x - 3)(x - 6) / (x - 3)(x - 3)
Notice that we have (x - 3) in both the numerator and denominator. This means we can cancel out these expressions, as long as x ≠ 3 (to avoid division by zero). By canceling out the common factor, we get:
(x - 6) / (x - 3)
Therefore, the simplified expression for all values of x for which it is defined is:
(x - 6) / (x - 3)
And the correct answer is option B: x - 6 / x - 3.
If you'd like to verify the answer yourself for any specific value of x, substitute that value into both the original expression and the simplified expression. They should yield the same result.