SOLUTION: Good evening ma'am or sir I don't know how to get the values of x here in this equation (clue: there are 3 answers, 1 fraction and 2 whole numbers), will you please help me. This i

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Good evening ma'am or sir I don't know how to get the values of x here in this equation (clue: there are 3 answers, 1 fraction and 2 whole numbers), will you please help me. This i      Log On


   

Question 114482: Good evening ma'am or sir I don't know how to get the values of x here in this equation (clue: there are 3 answers, 1 fraction and 2 whole numbers), will you please help me. This is the equation: x^2+2x+1/x^2+3x+2=x^2+5x+6/x^2+4x+3.
Thank you very much ma'am or sir.
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!
how to get the values of x here in this equation (clue: there are 3 answers, 1 fraction and 2 whole numbers), will you please help me. This is the equation:
%28x%5E2%2B2x%2B1%29%2F%28x%5E2%2B3x%2B2%29+=+%28x%5E2%2B5x%2B6%29%2F+%28x%5E2%2B4x%2B3%29
We will try to factor what ever is possible to factor in order to simplify this equation:
first this quadratic trinomial where we recognize the square of sum
x%5E2%2B2x%2B1+=++%28x%2B1%29%5E2+=+%28x%2B1%29%28x%2B1%29…………………….(1)
Now factor this one
x%5E2%2B3x%2B2…….we can replace 3x with 1x+%2B+2x

x%5E2%2B1x+%2B+2x+%2B2…….group first two terms and last two terms together

%28x%5E2%2B1x%29+%2B+%282x+%2B2%29…….notice that x is common for first group and 2 is common for second group; so factor them out
x%28x%2B1%29+%2B+2%28x+%2B1%29……. Factor out common x%2B1, combine like terms
+%28x%2B1%29+%28x+%2B+2%29+……………………………………………….(2)
Now factor this one
x%5E2+%2B+5x+%2B+6……….. replace 5x+with 2x%2B3x

x%5E2+%2B+2x+%2B+3x+%2B+6……….. group first two terms and last two terms together
%28x%5E2+%2B+2x+%29%2B%28+3x+%2B+6%29……….. factor a 1x out of the first group and factor a 3 out of the second group
x%28x+%2B+2+%29+%2B+3%28x+%2B+2%29……….. combine like terms

+%28x+%2B+2+%29+%28x+%2B+3%29+……………………………………………..(3)
Now factor this one

x%5E2%2B4x%2B3………………. replace 4x+with 1x%2B3x
x%5E2%2B1x%2B3x+%2B3……. group first two terms and last two terms together
%28x%5E2%2B1x%29+%2B+%283x+%2B3%29……. factor a 1x out of the first group and factor a 3 out of the second group
x%28x%2B1%29+%2B+3%28x+%2B+1%29……. combine like terms
+%28x%2B1%29+%28x+%2B+3%29+……………………………………………….(4)
Now substitute (1), (2), (3), and (4) in original equation:

%28x%5E2%2B2x%2B1%29%2F%28x%5E2%2B3x%2B2%29+=+%28x%5E2%2B5x%2B6%29%2F+%28x%5E2%2B4x%2B3%29
….. simplify
+%28x%2B1%29%2F+%28x+%2B+2%29+=++%28x+%2B+2+%29++%2F+%28x%2B1%29++….....here we have denominators x%2B2 and x%2B1; they cannot be equal to 0 ( to divide by 0 doesn't make sense
so we will find values of x for which it will be equal to 0

they would be equal to 0 if:
x%2B2+=+0
if x+=+-2.........................one solution}
and
x%2B1=0
if x=-1........................second solution}

now find third solution:

+%28x%2B1%29%28x%2B1%29+=++%28x+%2B+2+%29%28x+%2B+2%29+…..multiply
+x%5E2%2B+2x+%2B+1+=++x%5E2+%2B+4x+%2B+4+….....simplify (eliminate x%5E2 on both sides
+2x+%2B+1+=+4x+%2B+4+….....move 2x to the right and 4 to the left
1+-+4+=+4x+-2x+….....
-+3+=+2x+….....divide both sides by 2
-+3%2F2+=+2x%2F2+….....
-+3%2F2+=+x+….....

or
x+=+-3%2F2….......................third solution}