Equations and I...

If the roots of the equation bχ² + cχ + a = 0 is imaginary, then for all real values of χ, the expression 3b² χ² + 6bcχ + 2c² is

If a, b and c are in GP, then the equation aχ² + 2bχ + c = 0 and dχ² + 2eχ + f = 0 have a common root, if d/a, e/b and f/c are in

If ax² + bχ + c = 0 and 2χ² + 3χ + 4 = 0 have a common root, where a, b, c ∈ N (set of natural numbers), then the least value of a + b + c is

The quadratic equations χ² - 6χ + a = 0 χ² - cχ + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then, the common root is

Let a, b and c be real. If aχ² + bχ + c = 0 has two real roots α and β where α < - 1 and β > 1 , then

All the values of m for which both roots of the equation χ² - 2mχ + m² - 1 = 0 are greater than - 2 but less than 4 lie in the interval

The values of k, for which the equations χ² - kχ - 21 = 0 and χ² - 3kχ + 35 = 0 will have a common roots, is

If the both the root of the quadratic equation χ² - 2kχ + k² + k - 5 = 0 are less than 5, then k lies in the interval

If χ is real, then the value of