Discovering Real Numbers That Satisfy Complex Equations

In the realm of mathematics, solving equations with real numbers can be both challenging and rewarding. This article explores a specific problem involving complex equations and how to solve them. Let’s delve into the methods, insights, and applications of such equations.

Understanding the Problem

Suppose we want to find real numbers (a), (b), and (c) such that the following system of equations is satisfied:

$$begin{cases}a^2b^2c^2 1 ab sqrt{2}end{cases}$$

Step-by-Step Solution

Step 1: Simplify the Given Equations

The first equation can be written as:

$$a^2b^2c^2 1$$

And the second equation is:

$$ab sqrt{2}$$

Using the second equation, we can express (b) in terms of (a):

$$b frac{sqrt{2}}{a}$$

Step 2: Substitute and Simplify Further

Substitute (b frac{sqrt{2}}{a}) into the first equation:

$$a^2 left(frac{sqrt{2}}{a}right)^2 c^2 1$$

This simplifies to:

$$2c^2 1$$

And, hence:

$$c^2 frac{1}{2}$$

Therefore:

$$c pm frac{1}{sqrt{2}}$$

Step 3: Determine the Values of (a) and (b)

Now, let’s find the values of (a) and (b). We have:

$$ab sqrt{2}$$

And from the above, we know:

$$c^2 frac{1}{2}$$

Therefore, we can rewrite the equation as:

$$a^2 b^2 2 left(frac{1}{2}right) 1$$

This gives us two sets of solutions for (a) and (b):

1. (a frac{1}{sqrt{2}}, b frac{1}{sqrt{2}})

2. (a -frac{1}{sqrt{2}}, b -frac{1}{sqrt{2}})

Thus, the solutions for (a), (b), and (c) are:

$$left(a, b, cright) left(frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0right) text{ or } left(-frac{1}{sqrt{2}}, -frac{1}{sqrt{2}}, 0right)$$

Verification

To verify the solutions, substitute the values back into the original equations:

(a frac{1}{sqrt{2}}, b frac{1}{sqrt{2}}) (a^2 b^2 left(frac{1}{sqrt{2}}right)^2 left(frac{1}{sqrt{2}}right)^2 frac{1}{2} cdot frac{1}{2} frac{1}{4} cdot 2 1) (ab frac{1}{sqrt{2}} cdot frac{1}{sqrt{2}} frac{1}{2} sqrt{2}) (c 0)

The solutions ( left(-frac{1}{sqrt{2}}, -frac{1}{sqrt{2}}, 0right) )

follow the same verification process.

Conclusion

The problem of finding real numbers (a), (b), and (c) that satisfy the given equations has been solved by breaking it down into manageable steps. The key insights and methods used here are crucial for solving similar complex equations. The solutions are:

$$boxed{left(a, b, cright) left(frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0right) text{ or } left(-frac{1}{sqrt{2}}, -frac{1}{sqrt{2}}, 0right)}$$

Keywords

real numbers, complex equations, mathematical solutions