---
title: "Inverse z of (2z²-5z)/((z-2)(z-3))"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:inverse-z-2z2-5z"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/inverse-z-2z2-5z
site: BCT Engineering
---

# Inverse z of (2z²-5z)/((z-2)(z-3))

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2072 Asoj, 5 marks
- Video: [Solve This Z-Transform in 5 Minutes | Partial Fractions Method Explained](https://www.youtube.com/watch?v=QiNspeRxl70) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/inverse-z-2z2-5z
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the inverse z-transform of (2z² - 5z)/((z - 2)(z - 3)) by using partial fraction method.

## Short answer

Divide by z and use partial fractions: X(z)=z/(z-2)+z/(z-3). Inverse gives x(k)=2^k+3^k for k≥0.

## Answer

**Partial fractions.** Divide by z first, because every standard pair has a factor z in the numerator.

X(z)/z = (2z − 5)/[(z − 2)(z − 3)] = A/(z − 2) + B/(z − 3)

```text
A = (2·2 − 5)/(2 − 3) = (−1)/(−1) = 1
B = (2·3 − 5)/(3 − 2) = 1/1       = 1
```

So X(z) = z/(z − 2) + z/(z − 3).

Using Z⁻¹[z/(z − a)] = aᵏ:

**Answer:** x(k) = 2ᵏ + 3ᵏ, k ≥ 0.

Check: x(0) = 2 (= lim X(z)), x(1) = 5, x(2) = 13, x(3) = 35 — matches the power-series expansion of X(z).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. X(z)/z =
   A. (2z - 5)/[(z-2)(z-3)]
   B. (2z^2 - 5z)/(z-2)(z-3)
   C. (2z - 5)/z
   D. 1/[(z-2)(z-3)]
   Answer: A. Cancel z. (Key point: Divide by z)

2. A and B are
   A. A = 2, B = -5
   B. A = 1, B = 1
   C. A = -1, B = 1
   D. A = 1, B = -1
   Answer: B. Cover-up. (Key point: Coefficients)

3. Z inverse of z/(z-a) is
   A. a^-k
   B. k a^k
   C. a^k
   D. delta(k - a)
   Answer: C. Geometric. (Key point: Standard pair)

4. x(k) =
   A. 2^k - 3^k
   B. 3^k - 2^k
   C. 2^k * 3^k
   D. 2^k + 3^k
   Answer: D. Check x(0) = 2 = lim X(z). (Key point: Result)

## Close variants set in other exams

- [Inverse z of (z-4)/((z-1)(z-2)²)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ainverse-z-z-4-over-z-1-z-2-sq) — Applied Mathematics, chapter 2, asked 5×
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