Steps to determine the number and type of real solutions using the discriminant

📚 Understanding the Discriminant

The discriminant is a powerful tool derived from the quadratic formula that helps us determine the nature and number of real solutions of a quadratic equation without actually solving for those solutions. A quadratic equation is generally expressed in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. The discriminant, often denoted as $\Delta$ (delta), is given by the formula:

$\Delta = b^2 - 4ac$

📜 History and Background

The concept of the discriminant has been around for centuries, evolving alongside the development of algebra. Early mathematicians recognized that the solutions to quadratic equations could behave differently depending on the values of the coefficients. The formalization of the discriminant provided a systematic way to categorize these behaviors. It is a cornerstone in understanding the properties of quadratic equations and their solutions.

🔑 Key Principles

• 🔍 Definition: The discriminant ($\Delta$) is the expression $b^2 - 4ac$ derived from the quadratic equation $ax^2 + bx + c = 0$.
• 🔢 Calculating the Discriminant: Identify $a$, $b$, and $c$ from the quadratic equation and substitute them into the formula $\Delta = b^2 - 4ac$.
• 📊 Interpreting the Discriminant:
  • 📈 If $\Delta > 0$, the equation has two distinct real solutions.
  • 🤝 If $\Delta = 0$, the equation has exactly one real solution (a repeated root).
  • 📉 If $\Delta < 0$, the equation has no real solutions (two complex solutions).

💡 Steps to Determine the Number and Type of Real Solutions

1. 📝 Step 1: Write the quadratic equation in the standard form: $ax^2 + bx + c = 0$.
2. 🧮 Step 2: Identify the coefficients $a$, $b$, and $c$.
3. ➗ Step 3: Calculate the discriminant using the formula $\Delta = b^2 - 4ac$.
4. ✅ Step 4: Determine the number and type of real solutions based on the value of $\Delta$:
   • $\Delta > 0$: Two distinct real solutions.
   • $\Delta = 0$: One real solution (repeated root).
   • $\Delta < 0$: No real solutions (two complex solutions).

🌍 Real-World Examples

Example 1: Determine the number of real solutions for the quadratic equation $x^2 - 5x + 6 = 0$.

Here, $a = 1$, $b = -5$, and $c = 6$.

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

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

Example 2: Determine the number of real solutions for the quadratic equation $x^2 + 4x + 4 = 0$.

Here, $a = 1$, $b = 4$, and $c = 4.

$\Delta = (4)^2 - 4(1)(4) = 16 - 16 = 0$

Since $\Delta = 0$, the equation has one real solution (repeated root).

Example 3: Determine the number of real solutions for the quadratic equation $x^2 + x + 1 = 0$.

Here, $a = 1$, $b = 1$, and $c = 1$.

$\Delta = (1)^2 - 4(1)(1) = 1 - 4 = -3$

Since $\Delta < 0$, the equation has no real solutions (two complex solutions).

📝 Conclusion

Understanding the discriminant is crucial for quickly determining the nature of solutions to quadratic equations. By calculating $b^2 - 4ac$, you can easily identify whether an equation has two distinct real solutions, one real solution (repeated root), or no real solutions. This tool simplifies problem-solving and enhances your understanding of quadratic functions.