Solving the Mathematical Equation $sqrt{9x^{2} 12x 4} 1$

Diving into the world of algebra and solving equations with square roots can be quite an adventure. In this article, we'll explore the method to solve the equation $sqrt{9x^{2} 12x 4} 1$ step by step. Understanding this process can help in grasping the nuances of quadratic equations and their solutions, making it a valuable learning experience for students and professionals alike.

Squaring Both Sides of the Equation

To begin, we square both sides of the equation to eliminate the square root. The equation becomes:

x 2 12x 4 1 [/math>

But this is not quite the equation we started with. Upon closer inspection, the equation provided is:

9 x 2 mn>12x mn>4 mn>1 [/math]

Let's go through the steps:

9 x 2 mn>12x mn>4 mn>1 [/math] x 2 mn>12x mn>4 mn>1 [/math]

Now, we move all terms to one side to form a quadratic equation:

9 x 2 mn>12x mn>4 ? mn>1 mn>0 [/math]

This simplifies to:

x 2 mn>12x mn>3 mn>0 [/math]

Solving the Quadratic Equation

The quadratic equation can be factored as follows:

( x mn>1 ) ? ( x mn>3 ) mn>0 [/math]

Setting each factor to zero gives the solutions:

1. ( x 1 0 ) → ( x -1 )

2. ( x 3 0 ) → ( x -3 )

Checking Solutions

It’s important to check the solutions in the original equation to avoid extraneous solutions, especially when squaring both sides. Plugging the solutions back into the original equation:

For ( x -1 ):

9 (-1) 2 mn>12(-1) mn>4 9 - 12 4 mn>1 mn>1 [/math]

For ( x -3 ):

9 (-3) 2 mn>12(-3) mn>4 81 - 36 4 mn>49 mn>7 [/math]

Therefore, the only valid solution is ( x -1 ).

Conclusion

In conclusion, solving the mathematical equation $sqrt{9x^{2} 12x 4} 1$ involves squaring both sides, forming and solving a quadratic equation, and checking the solutions to ensure they are valid. The detailed steps and cautionary advice provided in this article can help in mastering similar types of algebraic problems.