Understanding the Equation: 64x² + 6400 − 100x² = 6400

Solving quadratic equations is a fundamental concept in algebra, and some equations—like the one 64x² + 6400 − 100x² = 6400—offer clear opportunities to explore simplification and problem-solving techniques. In this article, we’ll break down this equation, simplify it step-by-step, solve for x, and clarify common pitfalls. Whether you're a high school student tackling algebra or a lifelong learner brushing up your skills, this guide will help you master the process.

Understanding the Context

Breaking Down the Equation

The given equation is:
64x² + 6400 − 100x² = 6400

At first glance, the left-hand side combines both like terms (the x² terms) and a constant. To simplify, we begin by combining like terms.

64x^2 + 6400 - 100x^2 = 6400

Key Insights

Step 1: Combine Like Terms

We see two x² terms: 64x² and −100x². Adding these together:
64x² − 100x² = −36x²

So the equation becomes:
−36x² + 6400 = 6400

Notice that the constant 6400 appears on both sides. Subtracting 6400 from both sides eliminates unnecessary terms:
−36x² + 6400 − 6400 = 6400 − 6400
−36x² = 0

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The volume of the sphere is given by the formula:
V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left( \frac{a}{2} \right)^3 = \frac{4}{3} \pi \frac{a^3}{8} = \frac{\pi a^3}{6}
Now, the ratio of the volume of the sphere to the volume of the cube is:

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Final Thoughts

Step 2: Solve for x

Now divide both sides by −36:
x² = 0

Taking the square root of both sides gives:
x = 0

Why This Equation Has Only One Solution

The final result x = 0 reflects that this equation is a degenerate quadratic—it reduces to a linear equation after simplification. Quadratic equations typically yield two solutions due to the ± nature of square roots, but when the x² and x terms cancel out (or vanish), only a single solution remains. In this case, the dominant term is −36x², forcing x² to zero.

Key Takeaways

  • Combine like terms carefully before simplifying: always identify and group similar terms, especially x² and constants.
  • Recognize how coefficients affect the number and nature of solutions.
  • Simplify equations fully before solving — unnecessary terms obscure the path to the correct answer.
  • In this example, non-essential terms canceled completely, leading neatly to x = 0.