How much does the spring compress?

A 2.0 - kg mass is dropped 2.0 m above a spring with a spring constant 40.0 N/m. How much does the spring compress? Use g = 10 m/s².

Solution

Given Data:
m = 2.0kg
k = 40.0N/m
h = 2.0m
x = ?

Useful formulas:
Uₛ = ½kx²
Uₘ = mgh

Solution:
mg(h + x) = ½kx²
½kx² - mgx – mgh = 0

x² - (2mg/k)x – (2mg/k)h = 0

x² – 2 (mg/k)x + (mg/k)² = (mg/k)² + (2mg/k)h

(x - mg/k)² = (mg/k)² + (2mg/k)h

x - mg/k = ±√ {(mg/k)² + 2(mg/k)h}

x = mg/k±√ {(mg/k)² + 2(mg/k)h}

Calculation
mg/k = 2kg*10m/s ² / 40N/m = ½m

(mg/k)² = ¼m²

2(mg/k)h = 2·½m·2m = 2m²

(mg/k)² + 2(mg/k)h = ¼m² + 2m² = (⁹/₄)m²

√ {(mg/k)² + 2(mg/k)h} = (³/₂)m

x₁ = ½m + (³/₂)m = 2m
x₂ = - ½m - (³/₂)m = - 1m

Problem's answer: 2.0 m