Solving the Equation u³ – 5u² + 6u = 0: Step-by-Step Guide

If you're studying algebra or preparing for exams, solving polynomial equations like u³ – 5u² + 6u = 0 is a fundamental skill. This widely used cubic equation appears in various fields, including physics, engineering, and economics. In this article, we’ll walk through how to factor, solve, and interpret the roots of this equation using clear, beginner-friendly steps.

Understanding the Context

What is the Equation?

The equation to solve is:

u³ – 5u² + 6u = 0

At first glance, this cubic polynomial may seem complex, but it can be simplified using algebraic techniques.

Key Insights

Step 1: Factor Out the Common Term

Notice that each term on the left-hand side contains a u. Factoring out u gives:

u(u² – 5u + 6) = 0

This is the first key step — extracting the greatest common factor.

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Final Thoughts

Step 2: Factor the Quadratic Expression

Now focus on factoring the quadratic: u² – 5u + 6

Look for two numbers that multiply to 6 and add up to –5. These numbers are –2 and –3.

So,
u² – 5u + 6 = (u – 2)(u – 3)

Therefore, the equation becomes:

u(u – 2)(u – 3) = 0

Step 3: Apply the Zero Product Property

The Zero Product Property states that if a product of factors equals zero, at least one factor must be zero.