Describe and Correct the error

Question

k/k+5 + 5/k+3 = -2k/ k^2 +8k+15

LCD= (k+3)(k+5)

k(k+3)+5(k+5)=-2k

k^2+3k+5k+25=-2k

k^2+8k+25=-2k

k^2+10k+25=0

(k+5)^2=0

k=-5

Answer 1

The algebraic steps are all correct assuming that k/k+5 means k/(k+5), 5/k+3 means 5/(k+3), and the denominator of the term on the right side of the equation is enclosed in parentheses.  

 

However, if you plug in -5 for k in the original equation, we get a zero denominator.  So, -5 is an extraneous root that must be discarded.  Consequently, the equation has NO SOLUTION.  It's important to check your answers when solving equations in which a variable appears in a denominator! 

Answer 2

Being that we have a rational equation with unlike denominators, we need to find the use the LCD. 

 

We will go through the entire process to see if there are any mistakes here.

 

 

If we simplify the right side of the equation, the denominator on that side becomes (k + 3)(k + 5).  This is a common denominator on all sides.  So the LCD is correct.

 

Next we need to rewrite the equation using the LCD in a way that does not change the meaning of the original equation.

 

[k(k + 3) + k(k + 5)] /  (k + 3)(k + 5) = -2k / (k + 3)(k + 5)

 

 

We can equate the numerators.  Comparing this work to the one you have shown in your post, it matches, so you have this step correct.  So far so good.  

 

You eventually end up with

 

(k + 5)² = 0

 

k = -5

 

 

There is no mistake here algebraically, but always use parentheses and brackets to separate the terms.  It helps to make the equation easier and clearer to interpret.