Find two rational expressions that have a sum of 9-9x/x^2-9
I need steps not just answers
First, I assume you mean (9-9x)/(x^2-9)?
Making the same assumption as Ryan
9[(1-x)]/(x-3)(x+3)=9[a/(x-3)+b/(x+3)]
a(x+3) + b(x-3) = 1-x
a+b=-1
3a-3b=1
solve and get a =-1/3, b=-2/3
so
9 [(-1/3)/(x-3) -(2/3)/(x+3)]
= -3/(x-3) -6/(x+3)
If my assumption is true, you could simply use 9/(x^2-9) + -9x/(x^2+9), as these are both rational expressions, I think.
To find two rational expressions with a sum of 9-9x/x^2-9, you can follow these steps:
Step 1: Decompose the numerator and denominator of the sum expression (9-9x/x^2-9) into partial fractions.
The denominator (x^2-9) can be factored as (x+3)(x-3). Therefore, we can rewrite the expression as:
9-9x/x^2-9 = A/(x+3) + B/(x-3),
where A and B are constants to be determined.
Step 2: Determine the values of A and B by finding a common denominator and equating the numerators:
Multiplying both sides of the equation by the denominator (x+3)(x-3) gives:
9-9x = A(x-3) + B(x+3).
Step 3: Expand and simplify the right side of the equation:
Distributing A and B on the right side of the equation gives:
9-9x = Ax - 3A + Bx + 3B.
Step 4: Group the like terms on the right side of the equation:
Combining the x terms and the constant terms separately gives:
(-9x) = (Ax + Bx) + (-3A + 3B) = (A+B)x + (-3A + 3B).
Step 5: Equate the coefficients of x and the constant terms on both sides of the equation:
For the x terms, we have:
-9 = A + B.
For the constant terms, we have:
0 = -3A + 3B.
Step 6: Solve the system of equations obtained in step 5:
From the equation -9 = A + B, we can solve for A in terms of B:
A = -9 - B.
Substituting this value of A into the second equation:
0 = -3A + 3B,
0 = -3(-9 - B) + 3B,
0 = 27 + 3B + 3B,
0 = 6B + 27.
Solving this equation for B gives:
6B = -27,
B = -27/6,
B = -9/2.
Substituting this value of B into the equation A = -9 - B:
A = -9 + 9/2,
A = -18/2 + 9/2,
A = -9/2.
So, A = -9/2 and B = -9/2 are the values of A and B.
Step 7: Rewrite the original expression with the decomposed partial fractions:
Now that we have determined the values of A and B, we can write the original expression in the partial fraction decomposition form:
9-9x/x^2-9 = (-9/2)/(x+3) + (-9/2)/(x-3).
Therefore, two rational expressions that have a sum of 9-9x/x^2-9 are (-9/2)/(x+3) + (-9/2)/(x-3).