Let $a,b,c$ be distinct non-negative numbers and the vectors $a\widehat{i}+a\widehat{j}+c\widehat{k},\widehat{i}+\widehat{k},c\widehat{i}+c\widehat{j}+b\widehat{k}$ lie in the plane, then the quadratic equation $ax^2+2cx+b=0$ has

The condition that the three vectors lie in the same plane means their scalar triple product is zero. Calculating the determinant of the matrix formed by these vectors leads to the relation c^2 = ab. Substituting this into the quadratic equation ax^2 + 2cx + b = 0, the discriminant becomes (2c)^2 - 4ab = 4c^2 - 4ab = 4(ab) - 4ab = 0, which implies real and equal roots.