Solving the Equation √[x^2 2^x 6] 3 for x

In this article, we will explore how to solve the equation √[x^2 2^x 6] 3. This involves understanding the behavior of exponential functions, quadratic expressions, and how to manipulate and solve such equations. Let's break down the solution step by step.

Step-by-Step Solution

Starting with the given equation:

√[x^2 2^x 6] 3

First, square both sides to simplify the equation:

(√[x^2 2^x 6])^2 3^2

x^2 2^x 6 9

Subtract 9 from both sides to form a new equation:

x^2 2^x 6 - 9 0

x^2 2^x - 3 0

This equation is not straightforward to solve algebraically, as it contains both a quadratic term and an exponential term. However, we can use observation to find potential solutions.

Observation and Solution

By observation, consider x 1:

x^2 2^x - 3 1^2 2^1 - 3 1 2 - 3 0

Thus, x 1 is a solution.

Graphical Analysis

Graphing each side of the equation can also provide insight into the solution. The x-value at which the two sides of the equation intersect is the solution.

The graph of √[x^2 2^x 6] and the line y 3 intersect at x 1.63657603 and x 1.

Conclusion

Based on the algebraic and graphical analysis, we can conclude that:

x 1 is a solution.

There could be another solution between -1 and -2, approximately x ≈ -1.6, but further analysis or iterative methods would be required to confirm this.

In summary, solving the equation √[x^2 2^x 6] 3 involves algebraic manipulation and graphical verification. By squaring both sides and observing the behavior of the equation, we can identify the solution. For a more precise understanding, graphical methods and numerical analysis can be employed.

Keywords: solving equations, square root, exponential functions