lf the roots of the equation $ax^{2}+bx+c=0$ are the same , when each of the coefficient is increased by the same quantity, then $x^{2}+x+1$ is equal to

Let the roots of ax^2 + bx + c = 0 be equal to k. If we replace x with x + 1 in this equation, we get the new equation a(x + 1)^2 + b(x + 1) + c = 0, representing the coefficients increased by the same quantity. We can rewrite this as ax^2 + (2a + b)x + (a + b + c) = 0, where the new roots are k - 1. Since the roots are identical for both equations, k = k - 1, which means 1 = 0. This demonstrates that x^2 + x + 1 = 0.

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