Solving the Equation 1/4x * 4x 3: A Comprehensive Guide
Mathematics often involves solving various types of equations, including quadratic equations, which form a significant part of algebra. In this article, we will explore the solution to the equation 1/4x * 4x 3. This guide aims to provide a clear, step-by-step explanation of the process, making it accessible to students and enthusiasts alike.
Understanding the Equation
The given equation is 1/4x * 4x 3. It becomes apparent that if there is an X in the equation, it must be x. Let's break down the steps to solve this equation.
Simplifying the Equation
The equation 1/4x * 4x 3 simplifies to x^2 3. Here’s the breakdown:
1/4x * 4x simplifies to x^2 because the 4x and 1/4x cancel out the 4 and the x. The simplified equation is x^2 3.Solving the Simplified Equation
The simplified equation x^2 3 can be solved using the quadratic formula or by taking the square root of both sides.
x^2 3 can be rewritten as x ±√3.
The square root of 3 is approximately 1.732, and negative square root is -1.732.
Therefore, the solutions are:
x 1.732 or x -1.732
Further Simplification
Let's further breakdown the original equation and re-examine it:
1/4x * 4x 3 can be rewritten as 116x^2 12x.
Subtracting 12x from both sides, we get 16x^2 - 12x - 1 0.
We now have a quadratic equation in the form of ax^2 bx c 0, where a 16, b -12, and c -1.
Solving the Quadratic Equation
The quadratic formula is given by:
x (-b ± √(b^2 - 4ac)) / 2a
Substituting the values:
a 16
b -12
c -1
x (12 ± √((-12)^2 - 4 * 16 * -1)) / 2 * 16
x (12 ± √(144 64)) / 32
x (12 ± √80) / 32
x (12 ± 4√5) / 32
x (3 ± √5) / 8
This results in two solutions:
x (3 √5) / 8 ≈ 0.6545
x (3 - √5) / 8 ≈ 0.095491
Conclusion
In conclusion, solving the equation 1/4x * 4x 3 leads us to the solutions x (3 √5) / 8 and x (3 - √5) / 8. This guide has provided a detailed, step-by-step approach, making the process of solving quadratic equations accessible and straightforward.
Keywords: Quadratic equation, algebraic solution, equation simplification