If one root of the equation $\displaystyle (k^{2}+1)x^{2}+13x+4k=0$ is reciprocal of the other, then $k$ has the value

In a quadratic equation ax^2 + bx + c = 0, if one root is the reciprocal of the other, their product must equal 1. The product of the roots is given by c/a, so we set 4k / (k^2 + 1) equal to 1. Multiplying both sides by the denominator gives the quadratic equation k^2 - 4k + 1 = 0. Using the quadratic formula to solve for k yields k = [ 4 +/- sqrt(16 - 4) ] / 2, which simplifies to k = [ 4 +/- 2*sqrt(3) ] / 2. This provides two roots: 2 + sqrt(3) and 2 - sqrt(3), matching the required value of 2 - sqrt(3).

• Thanks
• Wrong explanation