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

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But the problem says achieves its minimum depth exactly once — so it must have one.
Therefore, the only way is if $ d(t) $ is not $ t^3 $, but a different cubic that fits the four points and has a unique minimum. But four points force $ d(t) = t^3 $.
Unless — the values are not $ d(1)=1 $, but rather $ d(1) = a(1)^3 + b(1)^2 + c(1) + d = 1 $, etc. — but that’s the same.

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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.)