Nature of Roots of a Quadratic Equation

A quadratic equation with integral coefficient has integral roots. Justify your answer.

The equation `3x^2 – 12x + (n – 5) = 0` has equal roots. Find the value of n.

If the quadratic equation px² − 2√5px + 15 = 0 has two equal roots then find the value of p.

Find the roots of the quadratic equation by using the quadratic formula in the following:

`1/2x^2 - sqrt(11)x + 1 = 0`

Find the values of k for which the roots are real and equal in each of the following equation:

(4 - k)x² + (2k + 4)x + 8k + 1 = 0

If one root of the quadratic equation x² + 12x – k = 0 is thrice the other root, then find the value of k.

Without solving, examine the nature of roots of the equation 4x² – 4x + 1 = 0

If x=−`1/2`, is a solution of the quadratic equation 3x²+2kx−3=0, find the value of k