Solving the Quadratic Equation: How to Solve 2x² + 2x – 144 = 0 Using Simplification (x² + x – 72 = 0)

Solve quadratic equations efficiently with a simple algebraic transformation. This article explores how converting the equation 2x² + 2x – 144 = 0 into a simpler form—x² + x – 72 = 0—makes solving for x much easier. Whether you’re a student, educator, or math enthusiast, this step-by-step guide will help you understand the logic and mechanics behind quadratic simplification for faster and clearer results.

Understanding the Context

Introduction to Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically expressed in the standard form:

ax² + bx + c = 0

Common examples include ax² + bx + c = 0, where a ≠ 0. These equations can yield two real solutions, one real solution, or complex roots, depending on the discriminant b² – 4ac. One key strategy to solving quadratics is completing the transformation to standard form—often by simplifying coefficients.

2x² + 2x – 144 = 0 → x² + x – 72 = 0

Key Insights

In this post, we focus on the effective trick of dividing the entire equation by 2 to reduce complexity: turning 2x² + 2x – 144 = 0 into x² + x – 72 = 0.

Step-by-Step Conversion: From 2x² + 2x – 144 = 0 to x² + x – 72 = 0

Original Equation:
2x² + 2x – 144 = 0

Divide all terms by 2 (to simplify coefficients):
(2x²)/2 + (2x)/2 – 144/2 = 0

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Question: A micropaleontologist studies a microfossil shaped like a hemisphere with radius $3x$ and a spherical fossil with radius $x$. What is the ratio of their volumes?
Solution: Volume of hemisphere: $\frac{2}{3}\pi (3x)^3 = \frac{2}{3}\pi \cdot 27x^3 = 18\pi x^3$.
Volume of sphere: $\frac{4}{3}\pi x^3$.

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Final Thoughts

Simplifies to:
x² + x – 72 = 0

Now the equation is simpler yet retains its core quadratic structure and key constants. The discriminant remains unchanged and easy to compute.

Why Simplify: The Benefits of Reformulating the Equation

Simplifying quadratic equations offers clear advantages:

  • Fewer coefficients to track: Smaller numbers reduce arithmetic errors.
  • Faster solution processing: Approaches like factoring, completing the square, or the quadratic formula become quicker.
  • Clearer insight: Recognizing patterns—such as factorable pairs—becomes more manageable.

In our case, dividing by 2 reveals that the equation relates directly to integers:
x² + x = 72 → (x² + x – 72 = 0)

This form perfectly sets up methods like factoring, quadratic formula, or estimation.

Solving the Simplified Equation: x² + x – 72 = 0