Understanding the Mathematical Expression: f(–3/2) = 4(–3/2 + 3/2)² = 0

Mathematics often reveals elegant simplicity beneath seemingly complex expressions. One such example is the function evaluation:
f(–3/2) = 4(–3/2 + 3/2)² = 0

At first glance, this equation may appear puzzling, but careful analysis uncovers its underlying logic. This article explains step-by-step how and why this expression simplifies to 0, showing the power of basic algebraic operations.

Understanding the Context

The Function: Step-by-Step Breakdown

The function f(x) at x = –3/2 is defined by the formula:

f(–3/2) = 4(–3/2 + 3/2)²

f\left(-\frac{3}{2}\right) = 4\left(-\frac{3}{2} + \frac{3}{2}\right)^2 = 4 \cdot 0 = 0

Key Insights

Let’s simplify the expression inside the parentheses:

–3/2 + 3/2 = 0

This is simply the sum of a number and its inverse on the number line—thingally zero.

Evaluating the Square

Continue Reading

We solve for $ k $: find the modular inverse of 7 modulo 13.
$ 7 \cdot 1 = 7
$ 7 \cdot 2 = 14 \equiv 1 \pmod{13} $

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Final Thoughts

Next, we square the result:

(–3/2 + 3/2)² = 0² = 0

Since any number squared equal to zero remains zero, this directly yields:

f(–3/2) = 4 × 0 = 0

Why This Matters

This example demonstrates a foundational principle in algebra: when an expression inside parentheses evaluates to zero, squaring it results in zero—then scaling by any real number (in this case, 4) preserves this property.

In broader terms, such evaluations help identify roots of functions, test symmetry, or simplify complex expressions in calculus and engineering.

Summary