Solve the partial fraction -1X -1/(X^2+2X+4)
To find the partial fraction decomposition of -1X - 1/(X^2 + 2X + 4), we can use the method of partial fractions.
First, let's factor the denominator X^2 + 2X + 4. The quadratic cannot be factored further, so we have:
X^2 + 2X + 4 = (X + 1)(X + 1)
Now, we can write the partial fraction decomposition as:
-1X - 1/(X^2 + 2X + 4) = A/(X + 1) + B/(X + 1)
We have the same factor (X + 1) repeated twice because the quadratic is not factorable.
Next, we need to determine the values of A and B. To do this, we can combine the fractions on the right side of the equation:
-1X - 1/(X^2 + 2X + 4) = (A + B)/(X + 1)
Since the numerators on both sides are equal, we have:
-1X - 1 = (A + B)(X + 1)
Expanding the right side gives:
-1X - 1 = AX + A + BX + B
Matching the coefficients of like terms:
-1X = AX + BX
-1 = A + B
This gives us a system of equations:
A + B = -1
A + B = 0
From the first equation, we can solve for A:
A = -1 - B
Substituting this into the second equation:
(-1 - B) + B = 0
-1 = 0
This equation is not true, which means there is no solution to the system of equations. Therefore, there is no proper partial fraction decomposition possible for -1X - 1/(X^2 + 2X + 4).