Nature of Roots and the Discriminant

The nature of the roots of a quadratic equation can be determined by examining the discriminant. For a quadratic equation of the form:

$$ ax^2 + bx + c = 0 $$

The discriminant is given by the formula:

$$ D = b^2 - 4ac $$

The value of \( D \) tells us about the nature of the roots:

  • If \( D > 0 \): The equation has two distinct real roots.
  • If \( D = 0 \): The equation has one real root (a repeated or double root).
  • If \( D < 0 \): The equation has two complex roots, which are conjugates of each other.

Example: Finding the Nature of Roots Using the Discriminant

Consider the equation:

$$ x^2 - 4x + 3 = 0 $$

Calculate the discriminant:

$$ D = (-4)^2 - 4(1)(3) = 16 - 12 = 4 $$

Since \( D > 0 \), this equation has two distinct real roots.

Practice Questions

  1. Determine the nature of the roots for the equation \( x^2 + 6x + 9 = 0 \).
    Solution

    Calculate the discriminant:

    $$ D = 6^2 - 4(1)(9) = 36 - 36 = 0 $$

    Since \( D = 0 \), the equation has one real root (a repeated root).

  2. Determine the nature of the roots for the equation \( 2x^2 + 3x + 5 = 0 \).
    Solution

    Calculate the discriminant:

    $$ D = 3^2 - 4(2)(5) = 9 - 40 = -31 $$

    Since \( D < 0 \), the equation has two complex roots.

  3. Determine the nature of the roots for the equation \( x^2 - 5x + 6 = 0 \).
    Solution

    Calculate the discriminant:

    $$ D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 $$

    Since \( D > 0 \), the equation has two distinct real roots.

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