Partial fractions of irreducible quadratic factors. Hi guys, today I will discuss partial fractions of irreducible quadratic factors. Have a look!!

If you are looking for more in partial fractions, do check in:

**How to get the partial fractions of lower degree numerators**

**Partial fractions of higher degree numerators**

**How to get the partial fractions of repeated roots**

**Partial fractions of equal degree expressions**

irreducible quadratic factors_compressed

**Solved examples of partial fractions of irreducible quadratic factors **

Now here comes my first example.

##### Example 1

Write the following expression into partial fraction form:

.

##### Solution

Now here the given expression is: . As I can see, I can’t factorize the denominator further. So this is an example of an expression with irreducible quadratic factor. Thus my task is to get the partial fraction of this expression.

**Step 1**

Also, I can rewrite this expression as

where and are constants.

Next, I’ll simplify the right-hand side of the equation. So this gives

Then I’ll simplify it a bit more to get

At the end, it becomes

Now the denominator of both the left-hand side and right-hand side are the same. So I can compare the numerators of both sides. And this gives

Next, I compare the coefficients of on both sides to get

(1)

Then I compare the coefficients of on both sides to get

(2)

Finally, I’ll compare the constants on both sides to get

(3)

Therefore I have three equations (1), (2) and (3) with three unknowns and . Now my task is to solve these equations to get the values of and .

**Step 2**

So I’ll substitute the value of equation (1) in equation (2) to get

And this means

Next, I’ll substitute this value of in equation (3) to get

So this means

Finally, I’ll substitute in equation (1) to get

So this means

Therefore I can say is the partial fraction form of .

And this is the solution to the given example.

Now I’ll give a few more examples.

Disclaimer: None of these examples is mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

**Example 2**

According to Stroud and Booth (2013)*, “Express each of the following in partial fractions: .”

**Solution**

Now here the given expression is: . As I can see, I can’t factorize the denominator further. So this is also an example of an expression with irreducible quadratic factor. Thus my task is to get the partial fraction of this expression.

**Step 1**

So I can rewrite this expression as

where and are constants.

Next, I’ll simplify the right-hand side of the equation. So this gives

Now I’ll separate the coefficients of and constants. So this gives

Then I’ll compare the coefficients on both sides. So this gives me three different equations.

First of all, I compare the coefficients of on both sides to get

(4)

Then I compare the coefficients of on both sides to get

(5)

In the end, I compare the constants on both sides to get

(6)

Therefore I have three equations (4), (5) and (6) with three unknowns and . Now my task is to solve these equations to get the values of and .

**Step 2**

So I’ll substitute from equation (4) to equation (5) to get

And this gives

Now I’ll substitute in equation (6) to get

So this means

Next, I substitute in equation (4) to get

Therefore I can say that is the partial fraction form of . And this is the answer to the given example.

Now I’ll give another example.

**Example 3**

According to Stroud and Booth (2013)*, “Express each of the following in partial fractions: .”

**Solution**

Now here the given expression is: . As I can see, I can’t factorize the denominator further. So this is also an example of an expression with irreducible quadratic factor. Thus my task is to get the partial fraction of this expression.

**Step 1**

So I can rewrite this expression as

where and are constants.

Next, I’ll simplify the right-hand side of the equation. So this gives

Now I’ll separate the coefficients of and constants. So this gives

Then I’ll compare the coefficients on both sides. So this gives me three different equations.

First of all, I compare the coefficients of on both sides to get

(7)

Then I compare the coefficients of on both sides to get

(8)

In the end, I compare the constants on both sides to get

(9)

Therefore I have three equations (7), (8) and (9) with three unknowns and . Now my task is to solve these equations to get the values of and .

**Step 2**

So from equation (9), I can say the value of is

Next, I’ll substitute this value of in equation (8) to get

(10)

Now I have got two different equations (7) and (10) in and . So, I’ll use these two equations to get the values of and .

Thus I can multiply equation (10) with and subtract it from equation (7) to get

Next, I’ll simplify it to get

Then I’ll put in equation (10) to get

Again I’ll substitute in equation (9) to get

Therefore I can say that is the partial fraction form of . And this is the answer to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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