---
title: "Inverse z of z/(z²+7z+10)"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:inverse-z-z-over-z2-7z-10"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/inverse-z-z-over-z2-7z-10
site: BCT Engineering
---

# Inverse z of z/(z²+7z+10)

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2071 Bhadra
- Video: [Inverse Z-Transform of z/(z^2 + 7z + 10) | VTU](https://www.youtube.com/watch?v=ZMsnwrTtHNQ) — 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-z-over-z2-7z-10
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the inverse z-transform of x(z) = z/(z² + 7z + 10).

## Short answer

Factor denominator and use partial fractions: X(z)= (1/3)[z/(z+2)-z/(z+5)]. Inverse gives x(k)=1/3[(-2)^k-(-5)^k] for k≥0.

## Answer

**Factorise.** z² + 7z + 10 = (z + 2)(z + 5).

X(z)/z = 1/[(z + 2)(z + 5)] = A/(z + 2) + B/(z + 5)

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

X(z) = (1/3) [ z/(z + 2) − z/(z + 5) ]

Using Z⁻¹[z/(z − a)] = aᵏ with a = −2 and a = −5:

**Answer:** x(k) = ⅓ [ (−2)ᵏ − (−5)ᵏ ], k ≥ 0.

Check: x(0) = 0, x(1) = 1, x(2) = −7, x(3) = 39. These also satisfy x(k) = −7x(k−1) − 10x(k−2), which comes from the denominator.

## Check yourself

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

1. z^2 + 7z + 10 =
   A. (z + 2)(z + 5)
   B. (z - 2)(z - 5)
   C. (z + 1)(z + 10)
   D. (z - 2)(z + 5)
   Answer: A. Roots -2 and -5. (Key point: Factorisation)

2. A and B are
   A. A = 1/3, B = 1/3
   B. A = 1/3, B = -1/3
   C. A = -1/3, B = 1/3
   D. A = 3, B = -3
   Answer: B. A = 1/(-2+5); B = 1/(-5+2). (Key point: Coefficients)

3. Roots a = -2, -5 give
   A. 2^k and 5^k
   B. (1/2)^k
   C. (-2)^k and (-5)^k
   D. k(-2)^k
   Answer: C. z/(z - a) with a negative. (Key point: Pairs)

4. x(k) =
   A. (1/3)[(-2)^k + (-5)^k]
   B. (1/3)[2^k - 5^k]
   C. (1/3)[(-5)^k - (-2)^k]
   D. (1/3)[(-2)^k - (-5)^k]
   Answer: D. x(1) = (-2+5)/3 = 1. (Key point: Result)

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