b^2 - 4ac = (-16)^2 - 4 \times 2 \times 30 = 256 - 240 = 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.

$b^2 - 4ac = (-16)^2 - 4 \times 2 \times 30 = 256 - 240 = 16$

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

🔗 Related Articles You Might Like:

📰 fort worth hotels 📰 wilderness lodge 📰 royal grove hotel waikiki beach 📰 Numbers That Are Not Rational 3507517 📰 Hot B A B E S 3294804 📰 From Charming Stranger To A List Sensation The Rise Of Young Pedro Pascal 8971044 📰 Insider Secrets How To Recover Your Locked 457 Account Fast 2983780 📰 When Can You Withdraw From 401K 3387502 📰 Master Microsoft Investing The Stock Calculator Thats Blowing Up Trading Results 3659295 📰 Unlock Free Anime Internet Games That Everyones Obsessing Over 9996435 📰 G7 Guitar Chord 2913844 📰 Master Histograms In Minutes The Ultimate Step By Step Guide You Need Now 1636939 📰 Youll Be Shocked Roth Ira Income Cap Revealedheres How To Qualify Instantly 2136362 📰 19 And Counting Family 8513420 📰 Will Johnson Nfl Draft 309000 📰 Firefox Download For Mac 9426945 📰 Peach State 3497000 📰 How A Body Piercing Navel Grants You The Ultimate Confidence Boost 9544135

Final Thoughts

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.)