64x^2 + 6400 - 100x^2 = 6400 - Coaching Toolbox
Understanding the Equation: 64x² + 6400 − 100x² = 6400
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.
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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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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.
