Reduce the following into lowest terms. Explain your solution.?

Question

1)

x^2 - 5x + 6
over
x^2 + x - 6

2)

x^2 - 9x + 20
over
25 -x^2

With solution please. And explanation.
please help me I really need some teaching help.

Answer 1

(X^2-5x+6)/(x^2+x-6)
=(x^2-3x-2x+6)/(x^2+3x-2x-6)
={x(x-3)-2(x-3)}/{x(x+3)-2(x+3)}
=(x-3)(x-2)/(x+3)(x-2)
=(x-3)/(x+3) [ after removing common x-2 from the numerator and the denominator]
2)x^2-9x+20
=x^2-4x-5x+20
=x(x-4)-5(x-4)
=(x-4)(x-5)
25-x^2
=(5)^2-(x)^2
=(5+x)(5-x)
= (x+5){-(x-5)}
=-(x+5)(x-5)
Therefore given expression
=(x-5)(x-4)/-(x+5)(x-5)
= - (x-4)/(x+5) [elliminating common factor x-5 from both numerator and denominator]

Answer 2

x^2 - 5x + 6 = (x - 2)(x - 3)

x^2 + x - 6 = (x - 2)(x + 3)
Because this is the denominator x cannot be one of 2, -3.

(x^2 - 5x + 6) / (x^2 +x - 6) = [(x - 2)(x - 3)] / [(x -2)(x + 3)] =

= (x - 3) / (x + 3)

2)

x^2 - 9x + 20 = (x - 4)(x + 5)

25 - x^2 = (5 - x)(x + 5) , x cannot be one of 5, -5

(x^2 - 9x + 20) / 25 - x^2 = [(x - 4)(x + 5)] / [(5 - x)(5 + x)] =
= (x - 4) / (5 - x)

Answer 3

1.

(x^2 - 5x + 6) / (x^2 + x -6)
==> factor numerator and denominator
(x - 2)(x - 3) / (x + 3)(x - 2)
==> cancel like factor of (x - 2) from top and bottom
(x - 3) / (x + 3) ... ANSWER

2.

(x^2 - 9x + 20) / (25 - x^2)
==> factor numerator and denominator
(x - 5)(x - 4) / (5 - x)(5 + x)
==> factor a negative sign from first term by reversing order*
-(5 - x)(x - 4) / (5 - x)(5 + x)
==> cancel out like factor of (5 - x) from top and bottom
-(x - 4) / (5 + x)
==> distribute negative sign by reversing order *
(4 - x) / (5 + x) ... ANSWER

* This rule is as follows: (x - y) = -(y - x), which was used in the labeled steps for the second problem

Answer 4

You just need to factor, and cancel out like terms in the numerator and denominator.

1.

(x-2)(x-3)/(x+3)(x-2)

=(x-3)/(x+3)

2.

(x-5)(x-4)/(5-x)(5+x)

=-(x-4)/(5+x)

=4+x/5+x


Ideas:

25-x^2 is a difference of squares, this equals (5-x)(5+x)
You can pull out a negative 1 for (x-5) and (5-x) because

-(-x+5)/(5-x)
-(5-x)/(5-x)=-1

Answer 5

1) Factor both top and bottom to get (x - 3)(x - 2) / (x + 3)(x - 2)
Cancel the common factor to get (x - 3) / (x + 3)

2) Factor both top and bottom to get (x - 5)(x - 4) / (5 - x)(5 + x)
Multiply on the bottom by -1 to get -(x - 5)(5 + x)
So now it looks like (x - 5)(x - 4) / -(x - 5)(5 + x)

Cancel the common factor, and remember to bring out the negative sign in the denominator to get -(x - 4) / (5 + x), which is more simply stated as (4 - x) / (5 + x)

Answer 6

1)
= (x^2 - 5x + 6) / (x^2 + x - 6)
= ([x- 2] [x - 3]) / ([x + 3] [x - 2])
= (x - 3) / (x + 3)

2)
= (x^2 - 9x + 20) / (25 - x^2)
= ([x - 5] [x - 4]) / ([- x - 5] [x - 5])
= (x - 4) / (- x - 5)