Question

Find the discriminant of the quadratic equation \( 4x^2 - 6x + 5 = 0 \) and hence find the nature of its roots.

Hint:

If the discriminant \( \Delta \) is negative, the quadratic equation has imaginary roots.

Answer 1

The given quadratic equation is: \[ 4x^2 - 6x + 5 = 0. \] The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula: \[ \Delta = b^2 - 4ac. \] For the given equation, \( a = 4 \), \( b = -6 \), and \( c = 5 \). Substituting these values into the discriminant formula: \[ \Delta = (-6)^2 - 4(4)(5) = 36 - 80 = -44. \] Since the discriminant is negative (\( \Delta = -44 \)), the roots of the quadratic equation are imaginary.
Conclusion:
The discriminant of the quadratic equation is \( -44 \), and hence the roots are imaginary.