The value of p for which equation $\displaystyle { 4x }^{ 2 }-2px+2p$ has equla root is

The equation is assumed to be 4x^2 - 2px + 2p = 0, and for it to have equal roots, the discriminant must equal zero. The discriminant is D = b^2 - 4ac = (-2p)^2 - 4(4)(2p) = 4p^2 - 32p. Setting the discriminant to zero gives 4p^2 - 32p = 0. Factoring out 4p results in 4p(p - 8) = 0, which yields p = 0 or p = 8; since p = 8 is the non-trivial solution given in the options, the value of p is 8.

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