3(x^2 - 2x + 3) = 3x^2 - 6x + 9 - Veritas Home Health
Understanding the Equation: 3(x² - 2x + 3) = 3x² - 6x + 9
Understanding the Equation: 3(x² - 2x + 3) = 3x² - 6x + 9
When studying algebra, one of the most fundamental skills is simplifying and verifying equations—a process that reinforces understanding of distributive properties, factoring, and polynomial expansion. A common example that demonstrates these principles is the equation:
3(x² - 2x + 3) = 3x² - 6x + 9
Understanding the Context
This seemingly simple equation serves as a perfect illustration of the distributive property and serves as a model for checking equivalence between expressions. In this SEO-optimized article, we’ll explore step-by-step how to expand, simplify, and verify this equation, providing clear insights for students, teachers, and self-learners seeking to improve their algebraic fluency.
What Is the Equation 3(x² - 2x + 3) = 3x² - 6x + 9?
At first glance, the equation presents a linear coefficient multiplied by a trinomial expression equal to a fully expanded quadratic trinomial. This form appears frequently in algebra when students learn how to simplify expressions and verify identities.
Key Insights
Left side: 3(x² - 2x + 3)
Right side: 3x² - 6x + 9
This equation is not just algebraic practice—it reinforces the distributive property:
a(b + c + d) = ab + ac + ad
Understanding this process helps students master more complex algebra, including factoring quadratic expressions and solving equations.
Step-by-Step Expansion: Why It Matters
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📰 Let $ A = x^2 + 4 $, $ B = y^2 + 2 $, and $ C = y^2 - 2 $, then: 📰 A^2 - 16BC = (A - 4\sqrt{BC})(A + 4\sqrt{BC}) 📰 But this factoring is not ideal over real numbers. Instead, observe that $ x^4 - 16y^4 = (x^2)^2 - (4y^2)^2 = (x^2 - 4y^2)(x^2 + 4y^2) $.Final Thoughts
To verify that both sides are equivalent, we begin with expansion:
Left Side:
3 × (x² - 2x + 3)
= 3×x² + 3×(-2x) + 3×3
= 3x² - 6x + 9
This matches the right side exactly.
Why This Step Is Crucial
Expanding ensures that both expressions represent the same polynomial value under every x. Misreading a sign or missing a coefficient can lead to incorrect conclusions—making expansion a foundational step in equation verification.
Verifying Equality: Confirming the Identity
Now that both sides expand to 3x² - 6x + 9, we conclude:
3(x² - 2x + 3) = 3x² - 6x + 9
✔️ True for all real values of x
This means the equation represents an identity—true not just for specific x-values, but across the entire real number line.
