Question

Find the discriminant of the quadratic equation \( 2x^2 - 4x + 3 = 0 \) and then find the nature of the roots.

Hint:

If the discriminant \( \Delta \) of a quadratic equation is negative, the roots are imaginary.

Answer 1

The given quadratic equation is: \[ 2x^2 - 4x + 3 = 0. \] The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ \Delta = b^2 - 4ac. \] For the equation \( 2x^2 - 4x + 3 = 0 \), we have \( a = 2 \), \( b = -4 \), and \( c = 3 \). Substituting these values into the discriminant formula: \[ \Delta = (-4)^2 - 4(2)(3) = 16 - 24 = -8. \] Since the discriminant is negative (\( \Delta = -8 \)), the roots of the quadratic equation are imaginary.
Conclusion:
The discriminant of the quadratic equation is \( -8 \), and hence the roots are imaginary.