Understanding the Quadratic Expression 8x² - 12x - 6x + 9: Simplification, Analysis, and Applications
When studying quadratic expressions, simplifying complex polynomials is a foundational skill that helps reveal their core characteristics and make them easier to analyze. One common expression encountered in algebra and calculus is:
8x² - 12x - 6x + 9
While it may look simple at first glance, understanding how to simplify and interpret this expression is essential for solving real-world problems, modeling quadratic scenarios, and preparing for advanced mathematical concepts.
Step 1: Simplify the Expression
The given expression is:
8x² - 12x - 6x + 9
Observe that —12x and -6x are like terms and can be combined:
- Combine the linear terms:
-12x – 6x = –18x
So, the expression simplifies to:
8x² – 18x + 9
This simplified quadratic is much easier to work with in subsequent steps.
What Is a Quadratic Expression?
A quadratic expression always takes the form ax² + bx + c, where:
- a, b, and c are constants (real numbers),
- a ≠ 0, ensuring the expression is genuinely quadratic (not linear).
For our simplified expression 8x² – 18x + 9, we identify:
- a = 8
- b = –18
- c = 9
Why Simplification Matters
Simplifying expressions:
- Reduces errors in calculations.
- Clarifies the function’s behavior.
- Facilitates graphing, solving equations, and identifying key features like roots, vertex, and axis of symmetry.
Key Features of the Quadratic Function
1. Vertex (Maximum or Minimum)
The vertex lies at the point (h, k), where:
- h = –b/(2a)
- k = f(h)
Compute h:
h = –(–18)/(2×8) = 18/16 = 9/8
Now compute k = f(9/8):
f(x) = 8x² – 18x + 9
f(9/8) = 8(81/64) – 18(9/8) + 9
= (648/64) – (162/8) + 9
= (81/8) – (81/4) + 9
Convert all to eighths:
= 81/8 – 162/8 + 72/8
= (81 – 162 + 72)/8 = –9/8
So, vertex = (9/8, –9/8) and since a > 0, the parabola opens upward — this is a minimum point.
2. Roots (Zeros of the Function)
To find x-intercepts, solve 8x² – 18x + 9 = 0
Use the quadratic formula:
x = [–b ± √(b² – 4ac)] / (2a)
Compute discriminant:
Δ = b² – 4ac = (–18)² – 4×8×9 = 324 – 288 = 36
Since Δ > 0, there are two real and distinct roots:
x = [18 ± √36]/(16)
x = [18 ± 6]/16
Thus:
x₁ = (18 + 6)/16 = 24/16 = 3/2
x₂ = (18 – 6)/16 = 12/16 = 3/4
Graphing and Applications
The graph of y = 8x² – 18x + 9 is a parabola:
- Opens upward (a > 0)
- Vertex at (1.125, –1.125)
- Passes through the x-axis at x = 3/4 and x = 3/2
- Symmetric about the vertical line x = 9/8 (the axis of symmetry)
This quadratic model can represent real-life scenarios such as:
- Projectile motion (height vs. time)
- Revenue/Z-shaped cost/profit functions
- Area calculations in geometric problems
Summary
- The original expression 8x² - 12x - 6x + 9 simplifies to 8x² – 18x + 9.
- Key features: axis at x = 9/8, vertex (9/8, –9/8), roots at x = 3/4 and x = 3/2.
- Understanding simplification and root-finding is crucial for solving quadratic equations and graphing.
- Applications span physics, economics, and geometry.
Mastering such expressions builds a strong foundation for solving advanced equations, optimizing functions, and interpreting mathematical models in both academic and real-world contexts.
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For further reading, explore how transformations affect parabolas and how quadratics are used in optimization and physics.
