b^2 - 4ac = (-16)^2 - 4 \times 2 \times 30 = 256 - 240 = 16 - Portal da Acústica
Understanding the Quadratic Formula: Solving b² – 4ac with Example (b² – 4ac = (−16)² − 4×2×30 = 16)
Understanding the Quadratic Formula: Solving b² – 4ac with Example (b² – 4ac = (−16)² − 4×2×30 = 16)
The quadratic formula is one of the most powerful tools in algebra, enabling students and professionals alike to solve quadratic equations efficiently. At its core, the formula helps determine the roots of a quadratic equation in the standard form:
ax² + bx + c = 0
Understanding the Context
Using this formula, the discriminant — a key component — is calculated as:
b² – 4ac
This value dictates the nature of the roots: real and distinct, real and repeated, or complex.
Key Insights
Decoding b² – 4ac with a Real Example
Let’s walk through a concrete example to clarify how this discriminant calculation works:
Given:
b² – 4ac = (−16)² – 4 × 2 × 30
First, calculate (−16)²:
(−16)² = 256
Next, compute 4 × 2 × 30:
4 × 2 × 30 = 240
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Now subtract:
256 – 240 = 16
👉 So, b² – 4ac = 16
Why This Matters: The Significance of the Discriminant
The discriminant (the expression under the square root in the quadratic formula) reveals vital information about the equation’s solutions:
- Positive discriminant (e.g., 16): Two distinct real roots exist.
- Zero discriminant: Exactly one real root (a repeated root).
- Negative discriminant: The roots are complex numbers.
In this case, since 16 > 0, we know the quadratic has two distinct real roots, and we can proceed to solve using the full quadratic formula:
x = [−b ± √(b² – 4ac)] / (2a)
(Note: Here, a = 2 — remember, the coefficient of x² influences the final solution.)
