Solving Quadratic Equations and Understanding Common Pitfalls

Quadratic equations are a common topic in algebra, and they often appear in various forms, from simple to more complex cases. Understanding the nuances in solving these equations is crucial, especially when it comes to avoiding common mistakes. Let's explore a typical quadratic equation problem and analyze a common error that can skew results.

The Problem Statement

Given the equations:

a2 - b2 48

a - b u221A48

a2 b 30

We need to find the values of a and b and the corresponding values of ab/3.

The Common Mistake and Its Resolution

In the provided solution, a common mistake was identified in the third step, where the term -2ab was omitted. This omission led to an incorrect transformation of the equation.

The Original Approach

The original steps can thus be re-examined to correct the equation transformation.

Starting with the equation:

a2 - b2 48 - 2ab

Transforming it to:

u0394a2 b2 - 2ab a2 b - 18

Which simplifies to:

48 - 2ab a2 b - 18

Now, let's address the solution provided.

The Corrected Approach

Substitute b 30 - a2 into the first equation:

a2 - (30 - a2)2 48 - 2a(30 - a2)

Simplifying this, we get:

a2 - (900 - 60a2 a4) 48 - 60a 2a3

Which simplifies to:

a4 - 2a3 - 60a2 60a - 435 0

This equation doesn't simplify easily and solving it using a symbolic solver like sympy will yield results. However, let's simplify further:

a4 - 2a3 - 60a2 60a - 435 0

sympy can provide the roots for this polynomial.

Alternative Solutions and Verification

Let's consider an alternative solution where a - b -u221A48. The equations become:

a2 b2 48 2ab

30 - a2 b u221A48

a2 (30 - a2) 30

Solving these equations, we can substitute and simplify to find the values of ab/3.

a2 30 - a2 30

a2 (30 - a2) 30

Which simplifies to:

30 - a2 -u221A48

And solving it gives:

a u221A30 - u221A48 or a 1.598

Thus, ab/3 2.840.

Conclusion

Understanding the nuances in solving quadratic equations and recognizing common pitfalls can significantly impact the accuracy of your results. Paying close attention to algebraic manipulations and verifying solutions are key to achieving correct outcomes. Whether using manual methods or symbolic solvers, it’s important to double-check every step to ensure accuracy.