Find the nature of the roots of the following quadratic equations:

2x² – 8x + 5 = 0

Find the nature of the roots of the following quadratic equations:

3x² –2√6x + 2 = 0

Find the nature of the roots of the following quadratic equations:

5x² – 4x + 1 = 0

Find the nature of the roots of the following quadratic equations:

5x (x – 2) + 6 = 0

Find the nature of the roots of the following quadratic equations:

12x² – 4√15x + 5 = 0

Find the nature of the roots of the following quadratic equations:

x² – x + 2 = 0

If a and b are distinct real numbers, show that the quadratic equation 2 (a² + b²)x² + 2(a + b)x + 1 = 0 has no real roots.

Show that the roots of the equation x² + px – q² = 0 are real for all real values of p and q.

For what values of k are the roots of the quadratic equation 3x² + 2kx + 27 = 0 real and equal?

For what value of k are the roots of the quadratic equation kx(x –2√5) + 10 = 0 real and equal?

For what values of p are the roots of the equation 4 x² + px + 3 = 0 real and equal?

Find the nonzero value of k for which the roots of the quadratic equation 9x² – 3kx + k = 0 are real and equal.

Find the values of k for which the quadratic equation (3k + 1) x² + 2(k + 1)x + 1 = 0 has real and equal roots.

Find the values of p for which the quadratic equation (2p + 1)x² – (7p + 2)x + (7p – 3) = 0 has real and equal roots.

Find the values of p for which the quadratic equation (p + 1)x² – 6 (p + 1) x + 3 (p + 9) = 0, p 1 has equal roots. Hence, find the roots of the equation.

If – 5 is a root of the quadratic equation 2x² + px – 15 = 0 and the quadratic equation p (x² + x) + k = 0 has equal roots, find the value of k.

If 3 is a root of the quadratic equation x² – x + k = 0, find the value of p so that the roots of the equation x² + k (2x + k + 2) + p = 0 are equal.

If – 4 is a root of the equation x² + 2x + 4p = 0, find the value of k for which the quadratic equation x² + px (1 + 3k) + 7(3 + 2k) = 0 has equal roots.

If the quadratic equation (1 + m²)x² + 2mcx + c² – a² = 0 has equal roots, prove that c² = a²(1 + m²) .

If the roots of the equation (c² – ab)x² – 2(a² – bc)x + (b² – ac) = 0 are real and equal, show that either a = 0 or (a³ + b³ + c3) = 3abc.

Find the values of p for which the quadratic equation 2x² + px + 8 = 0 has real roots.

Find the value of a for which the equation (a – 12)x² + 2(a – 12)x + 2 = 0 has equal roots.

Find the value of k for which the roots of 9x² + 8kx + 16 = 0 are real and equal.

Find the values of k for which the given quadratic equation has real and distinct roots:

If a and b are real and ab then show that the roots of the equation (a – b)x² + 5(a + b)x – 2(a – b) = 0 are real and unequal.

If the roots of the equation (a² + b²)x² – 2 (ac + bd)x + (c² + d²) = 0 are a c equal, prove that

If the roots of the equations ax² + 2bx + c = 0 and are simultaneously real then prove that b² = ac.