The discriminant is a pivotal concept in the study of quadratic equations. It not only reveals the nature of the roots but also their quantity, thus enriching our understanding of these equations.

Discriminant of a Quadratic Expression

The discriminant relates to the roots of a quadratic equation in standard form $ax^2 + bx + c = 0.$

Definition:

The discriminant of the quadratic equation $ax^2 + bx + c$ is denoted and calculated as: $\Delta = b^2 - 4ac$

Interpretation of the Discriminant

The value of the discriminant provides crucial information about the roots of the quadratic equation:

• Positive Discriminant ( \Delta > 0 ):

  • Indicates two distinct real roots.
  • The roots are real and different because the square root of a positive number is real and distinct.

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• Zero Discriminant $( \Delta = 0 )$:

  • Indicates one real root, repeated twice.
  • This situation arises because the square root of zero is zero, leading to a single, repeated solution.

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• Negative Discriminant ( \Delta < 0 ):

  • Indicates no real roots.
  • The square root of a negative number is not real, hence no real solutions exist.

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Example 1:

For the equation $3x^2 - 6x + 2 = 0$, determine the nature of the roots.

Solution:

Since \Delta > 0, the equation has two distinct real roots.

Examples 2:

Find the condition for $m$ in the equation $x^2 - (m + 3)x + m = 0$ to have real and equal roots.

Solution:

For real, equal roots, $\Delta = 0 \$

$m^2 + 2m + 9 = 0$

Thus, $m$ must satisfy the equation $m^2 + 2m + 9 = 0$ for the roots to be real and equal.

Example 3:

Determine whether the equation $5x^2 + 2x + 1 = 0$ has any real roots.

Solution:

Since \Delta < 0, the equation has no real roots.