Root counts and a parameter

For ax2+bx+c=0, first require a 0. Then D=b2-4ac gives two distinct real roots if D>0, one repeated root if D=0, and no real root if D<0. The formula contains +- D: positive D gives different real numbers, zero makes them equal, and negative D has no real square root. Identify the actual coefficients; find any parameter that makes a=0 and solve that case separately; calculate D; solve the required condition; combine only the relevant cases. Example For (k-2)x2+x+1=0, k 2, D=9-4k. A repeated root requires k= 94. At k=2, the equation is instead x+1=0, also one real solution but not a repeated quadratic root. Exam check: “one real solution” may include a linear case; “equal roots of a quadratic” does not.