Determine the value of k for which the equation 4x^2+kx+4=0 has one root equal to double the other root.

Expert Answers
justaguide eNotes educator| Certified Educator

Let the value of one of the roots be R, the other root is 2R.

Now the quadratic equation can be derived by the following operation: (x - R)(x - 2R) = 0

=> x^2 - xR - 2xR + 2R^2 = 0

=> x^2 - 3xR + 2R^2 = 0

This is equivalent to 4x^2 + kx + 4 = 0

Dividing this by 4

=> x^2 + (k/4)x + 1 = 0

Now equate the coefficients of the x^2 - 3xR + 2R^2 = 0 and x^2 + (k/4)x + 1 = 0

We get k/4 = -3R and 2R^2 = 1

2R^2 = 1

=> R^2 = 1/2

=> R = 1/ sqrt 2 or -1/ sqrt 2

substitute this in k/4 = -3R

=> k = -12*(1/sqrt 2) or k = 12/sqrt 2

=> k = 6 sqrt 2 or k = -6 sqrt 2

Therefore k can take the values k = 6 sqrt 2 and k= -6 sqrt 2

airhead996 | Student

For k = -6, one of the roots is not double the other root. Instead we get .75 + .66i and .75 - .66i as two complex roots.