u^3 - 5u^2 + 6u = 0
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.
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.
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.
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.
Set each factor equal to zero:
- u = 0
- u – 2 = 0 → u = 2
- u – 3 = 0 → u = 3
Final Solution
The solutions to u³ – 5u² + 6u = 0 are:
u = 0, u = 2, u = 3
These are the three real roots of the cubic equation.
Why Are These Roots Important?
- u = 0 represents a point where the function crosses the u-axis at the origin.
- u = 2 and u = 3 are the other intercepts, useful in modeling growth, decay, or equilibrium points.
- In applied contexts, such equations model scenarios like profit and loss, velocity changes, or system dynamics.
Summary
- Factoring helped reduce the cubic equation into simpler linear terms.
- The zero product property enabled easy root identification.
- The roots u = 0, 2, and 3 provide full solution insight.
FAQ: Common Questions About u³ – 5u² + 6u = 0
Q: Why can’t we solve this by trying to “u³ – 5u² + 6u = 0” directly? A: Factoring makes solutions simpler and more transparent than trial-and-error methods. Polynomial factoring is a foundational algebra skill.
Q: What kind of equation is this? A: It’s a cubic equation (degree 3), where the highest power of u is 3.
Q: How are these roots used in real life? A: Roots of polynomials can identify key values in physics (e.g., time when velocity is zero), in optimization problems, or in financial models.
Q: Can this equation have complex roots? A: No. Since all coefficients are real and it factors completely into real linear terms, all three roots are real numbers.
Want to Practice More?
Try solving similar equations like:
- u³ – 4u² – u + 4 = 0
- 2u³ + 3u² – 2u = 0
Use factoring, the Rational Root Theorem, or synthetic division if needed.
Conclusion
The equation u³ – 5u² + 6u = 0 may look intimidating at first, but with factoring and the zero product property, it becomes straightforward. Mastering such equations strengthens your algebra foundation and prepares you for advanced mathematical and scientific challenges. Keep practicing — each polynomial solved deepens your understanding!
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