Help solve this confusing question asap please

Question

Solve

 4k-1 - k+1 = k²-4k+4

 k+2    k-2        k²-4

 

Thanks so much!

Answer 1

I believe this is a rational equation.   This means that we have fractional terms.  First, we factor the terms in the equation where possible.

 

[(4k - 1) / (k + 2)] - [(k + 1) / (k - 2)] = (k² - 4k + 4) / (k² - 4)

 

 

The right side of equation of can be factored.  On the right side,

 

(k - 2)(k - 2) / (k - 2)(k + 2) = (k - 2) / (k + 2)

 

 

The equation so far is

 

[(4k - 1) / (k + 2)] - [(k + 1) / (k - 2)] = (k - 2) / (k + 2)

 

 

The LCD is  (k + 2)(k - 2).

 

[(4k - 1)(k - 2)] - [(k + 1)(k + 2)] / (k + 2)(k - 2) = [(k - 2)(k - 2)] / (k + 2)(k - 2)

 

 

Now that we have same denominators, we only need to equate numerators.

 

(4k - 1)(k - 2) - (k + 1)(k + 2) = (k - 2)(k - 2)

 

 

Expand to get rid of parentheses.

 

4k² - 9k + 2 - (k² + 3k + 2) = k² - 4k + 4

 

4k² - 9k + 2 - k² - 3k - 2 = k² - 4k + 4

 

3k² - 12k = k² - 4k + 4

 

 

Move all terms to the left side to make the right side equal to zero.

 

2k² - 8k - 4 = 0

 

2(k² - 4k - 2) = 0

 

 

We have a quadratic equation.  Set the term in parenthesis equal to zero.

 

k² - 4k - 2 = 0

 

 

Use the quadratic formula to find the values of k.

 

k = (4 ± √(16 - 4(-2))) / 2

k = (4 ± √(16 + 8)) / 2

k = (4 ± √(24)) / 2

 

k = (4 ± 4.90) / 2

 

k = (4 + 4.90) / 2                 and                k = (8 - 4.90) / 2

 

k = 4.45                               and                k = -0.45

 

 

These are our solutions for k.

 

Next thing for you to do is the verify which value is accepted, by plugging them into the equation.  The left side of left side of equation must be equal in order for the values to be accepted as the solution.