Find the value of k

Question

Find the value of k if the roots of x²-4x-kx+49=0 are equal.

Answer 1

We want the two roots to be equal. This means it is a perfect square. For that, we look at the free coefficient, 49. It is square of 7 or -7.

We evaluate (x-7)².

 

(x-7)²=x²-14x+49.

 

In order for two polynomials to be equal, their corresponding coefficient must be equal.

In our case,

-14 = -4-k

-14+4 = -k

k=10.

 

Using the same argument, we find the other one

14 = -4 - k

14 +4 = -k

k = -18.

 

Any questions?

Answer 2

x²-(4+k)x+49=0   Set the discriminant to zero.

 

b²-4ac=[-(4+k)]²-4(1)49=0 --> (4+k)²=4(49)  (take sqrt)

4+k=±2(7)=±14  so k=±14-4 

 

so k=14-4=10   or   k=-14-4=-18

 

You can check this by plugging back in and factoring:

x²-14x+49=0      and        x²+14x+49=0