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Solving the Quadratic Equation for y4x^2-x-1
Solving the Quadratic Equation for y4x^2-x-1
Quadratic equations are a fundamental part of algebra, finding applications in a wide range of fields, from physics to engineering. One such equation is given by y 4x2 - x - 1. Solving this equation for x in terms of y involves a few steps, which are outlined in this article. We will guide you through the process, ensuring clarity and precision in understanding and solving the equation.
Understanding the Equation
The equation y 4x2 - x - 1 is a standard form of a quadratic equation. Here, y is a function of x, and we are interested in expressing x in terms of y. This type of problem often arises in various practical situations where you need to find the value of one variable (x) given the value of another (y).
Solving for x in Terms of y
To solve the equation y 4x2 - x - 1 for x in terms of y, follow these steps:
Start with the given equation:y 4x2 - x - 1
Move all terms to one side to form a quadratic equation:4x2 - x - 1 - y 0
Identify the coefficients:Here, a 4, b -1, and c -1 - y
Apply the quadratic formula:The general form of the quadratic formula is:
x [-b ± √(b2 - 4ac)] / (2a)
Substituting the values of a, b, and c into the formula gives:
x [1 ± √((-1)2 - 4 times; 4 times; (-1 - y))] / (2 times; 4)
Simplify the expression:x [1 ± √(1 16 16y)] / 8
x [1 ± √(16y 17)] / 8
Final Expression
The final expression for x in terms of y is:
x [1 ± √(16y 17)] / 8
This gives two possible solutions for x, one using the plus sign and the other using the minus sign. The choice of sign depends on the specific value of y and the desired solution.
Practical Applications
Solving quadratic equations is not just an abstract mathematical exercise. It has numerous practical applications:
In physics, it can be used to model the motion of objects under gravity.
In engineering, it can be used to design structures and calculate forces.
In economics, it can be used to model supply and demand functions.
In computer science, it can be used in algorithms and simulations.
Conclusion
Understanding how to solve the quadratic equation y 4x2 - x - 1 is not just a mathematical exercise but a skill with wide-ranging applications. By mastering the process, you can apply it to various real-world scenarios, making it a valuable tool in your problem-solving arsenal.
References
For further reading and deeper understanding, consider exploring advanced textbooks on algebra, calculus, and applied mathematics.
Keywords: quadratic equation, solving for x, algebraic expression