t(t^2 - 5t + 6) = 0 - Silent Sales Machine

Understanding the Equation t(t² - 5t + 6) = 0: A Complete Guide

Solving polynomial equations is a fundamental skill in algebra, and one particularly interesting expression is t(t² - 5t + 6) = 0. This equation combines linear and quadratic components, making it a great example for learning factoring, zero-product property, and quadratic solutions. In this article, we’ll explore how to solve this equation step-by-step, interpret its roots, and understand its applications.

Understanding the Context

What Does t(t² - 5t + 6) = 0 Mean?

The equation t(t² - 5t + 6) = 0 is a product of two factors set equal to zero. According to the Zero-Product Property, if the product of two factors equals zero, then at least one of the factors must be zero. So we set each factor equal to zero:

First factor: t = 0
Second factor: t² - 5t + 6 = 0

Solving these parts separately will give us all the solutions to the original equation.

Key Insights

Step 1: Solve the Linear Factor t = 0

This is straightforward:
t = 0 is a solution by itself.

Step 2: Solve the Quadratic t² - 5t + 6 = 0

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Final Thoughts

The quadratic equation t² - 5t + 6 = 0 can be solved by factoring, completing the square, or using the quadratic formula. Factoring works cleanly here.

Factoring

We look for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3.
So,
t² - 5t + 6 = (t - 2)(t - 3) = 0

Setting each factor to zero gives:
t - 2 = 0 → t = 2
t - 3 = 0 → t = 3

Final Solutions

Combining both parts, the full set of solutions to t(t² - 5t + 6) = 0 is:

t = 0
t = 2
t = 3

Thus, the roots are t = 0, 2, and 3. These three real and distinct solutions come from combining one linear and one quadratic factor.