64x^2 + 6400 - 100x^2 = 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.
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.
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
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.
Practical Applications
While this particular equation simplifies down to a single value, understanding how to manipulate and solve equations like 64x² + 6400 − 100x² = 6400 helps in real-world problem-solving — from physics models to financial calculations — where simplifying expressions is essential.
FAQ: Common Questions About This Type of Equation
Q: Why does -36x² = 0 give only one solution? A: Because the equation defines x² as zero. The only real number whose square is zero is zero itself.
Q: Could there be complex solutions here? A: Technically, yes—since x² = 0 still gives x = 0 (with multiplicity two), but in real numbers, only x = 0 is valid.
Q: When does this type of simplification occur often? A: Whenever constant terms dominate simplifications or when coefficients combine to nullify higher-degree terms, as with 64x² − 100x² here.
Final Thoughts
The equation 64x² + 6400 − 100x² = 6400 may seem simple, but mastering its solution teaches powerful algebraic skills: combining terms accurately, recognizing patterns, and confidently solving up to quadratic forms. Always simplify carefully, verify each step, and trust the logic of the equation — skills that serve far beyond solving one equation.
If you found this explanation helpful, share it with fellow learners, and stay curious with algebra — every equation is a puzzle waiting to be solved!
Related Topics: Quadratic Equations, Algebraic Simplification, Solving Variable Equations, Solving x² = 0, Practical Algebra Examples
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