Discriminant Analysis

Introduction and Purpose

The discriminant: $\Delta = b^2 - 4ac$ is a key mathematical concept used to determine the type of roots in quadratic equations. Grasping its importance is vital for recognising:

• Real and distinct roots
• One real repeated root
• Complex conjugate roots

Discriminant Importance

• Assists in identifying the type and quantity of solutions.
• Crucial in physics and engineering for addressing complex real-world scenarios.

Role in Determining Roots

• Nature of Roots based on the discriminant $\Delta$:
  • $\Delta > 0$: Two distinct real roots.
  • $\Delta = 0$: One real repeated root.
  • $\Delta < 0$: Two complex conjugate roots.

Visual Learning Enhancements

• Utilise diagrams for educational purposes:

  • Parabolas intersecting the x-axis to demonstrate the discriminant's impact.

  

Calculation and Baseline Understanding

Example Problems

• Calculating $\Delta$
  • Example: $x^2 - 4x + 3 = 0$
    • Identify coefficients: $a = 1$, $b = -4$, $c = 3$.
    • Calculate: $(-4)^2 - 4 \times 1 \times 3 = 16 - 12 = 4$.
    • Two distinct real roots since $\Delta = 4 > 0$.
• Additional Examples:
  • $x^2 - 2x + 1 = 0$: $\Delta = 0$ (one repeated root).
  • $x^2 + x + 1 = 0$: $\Delta = -3$ (complex roots).

Worked Examples

1. Case $\Delta > 0$:

   • Example: $x^2 - 5x + 6 = 0$
     • First, identify the coefficients: $a = 1$, $b = -5$, $c = 6$
     • Calculate the discriminant: $\Delta = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$
     • Since $\Delta > 0$, there are two distinct real roots
     • Using the quadratic formula: $x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}$
     • Therefore, roots are $x = 2$ and $x = 3$

2. Case $\Delta = 0$:

   • Example: $x^2 - 4x + 4 = 0$
     • First, identify the coefficients: $a = 1$, $b = -4$, $c = 4$
     • Calculate the discriminant: $\Delta = (-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0$
     • Since $\Delta = 0$, there is one repeated root
     • Using the quadratic formula: $x = \frac{4}{2} = 2$
     • Therefore, the repeated root is $x = 2$

3. Case $\Delta < 0$:

   • Example: $x^2 + x + 1 = 0$
     • First, identify the coefficients: $a = 1$, $b = 1$, $c = 1$
     • Calculate the discriminant: $\Delta = 1^2 - 4 \times 1 \times 1 = 1 - 4 = -3$
     • Since $\Delta < 0$, there are complex conjugate roots
     • Using the quadratic formula: $x = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}$
     • Therefore, the roots are complex conjugates

Common Misconceptions

• Error Sources: Incorrect calculation or misinterpretation of the discriminant.

Practice Problems with Solutions

1. Compute $\Delta$ for $x^2 - 7x + 10 = 0$.

   • Solution: $a = 1$, $b = -7$, $c = 10$
   • $\Delta = (-7)^2 - 4 \times 1 \times 10 = 49 - 40 = 9$
   • Since $\Delta > 0$, there are two distinct real roots.

2. Determine $\Delta$ for $x^2 - 6x + 9 = 0$.

   • Solution: $a = 1$, $b = -6$, $c = 9$
   • $\Delta = (-6)^2 - 4 \times 1 \times 9 = 36 - 36 = 0$
   • Since $\Delta = 0$, there is one repeated root, which is $x = 3$.

3. Evaluate $\Delta$ for $x^2 + x + 2 = 0$.

   • Solution: $a = 1$, $b = 1$, $c = 2$
   • $\Delta = 1^2 - 4 \times 1 \times 2 = 1 - 8 = -7$
   • Since $\Delta < 0$, there are complex conjugate roots.

Application Examples

• Projectile Paths: Employ discriminants to predict intersections with targets.
  • Outcome: Lack of real intersection results in missing the target.
• Engineering: Apply checks in evaluating material stress for project viability.

Conclusion

Comprehending discriminants facilitates the prediction of root types, enabling thorough preparation for exams and practical applications in disciplines such as physics and engineering.