Solving and Understanding Square Equations: A Comprehensive Guide

Understanding and solving square equations is a fundamental part of algebra. This article explains how to solve a specific square equation: the square of a number is three times the number itself. We will explore the equation X_squared2X 3 and provide a step-by-step solution along with a detailed explanation.

The Equation: X_squared2X 3

The equation we are given is X_squared2X 3. This can be written as:

X_squared - 2X - 3 0

Solving the Equation

Start by writing the equation in standard form:

X_squared - 2X - 3 0

Next, we can factor this quadratic equation to find the roots:

X_squared - 3X X - 3 0

(This is broken down for clarity, but we can directly use the factors without further expansion)

The equation can be factored as:

X-30 or X 10

Solving these two linear equations:

X3 or X-1

Therefore, X3 or X-1.

Proofs and Verification

To verify, let's substitute X3 into the original equation:

3_squared 2(3) 3 9 (which is true)

Similarly, substituting X-1:

(-1)_squared 2(-1) 3 1 (which is also true)

Generalization and Further Considerations

The equation X_squared - 2X - 3 0 can be solved in many ways, including the quadratic formula, factoring, and completing the square. The key is to understand the algebraic manipulations and verify each solution.

Let's consider a similar problem: the square of a number is 3 more than twice the number itself. In this context, the equation is X_squared - 2X - 3 0. The solutions are X3 or X-1.

Conclusion and Final Thoughts

Solving and understanding square equations is crucial in algebra and has numerous applications in various fields, from physics to engineering. By breaking down the problem step-by-step, we can solve complex equations and ensure our solutions are correct.

Remember, the key is to practice and understand the underlying concepts thoroughly.