Solve the partial fraction -1X -1/(X^2+2X+4)

Answer 1

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).