Roots of f2x 0 and f(x) 0

In this article, we will explore the relationship between the roots of the quadratic equation f(x) 0 and the equation f2x 0. We will provide a detailed explanation and derivation to help you understand how to find the roots of f2x 0 given the roots of f(x) 0.

Understanding the Equations

Let's consider the function f(x) to be a quadratic equation. For simplicity, let's assume that the roots of f(x) 0 are given as 1 and -2. To proceed, we first define the equation f(x) 2x - 12x^2 0.

Step 1: Simplifying the Equation

To simplify the given equation, we first express it in a standard form:

2x - 12x^2 0

This can be factored as:

2x(1 - 6x) 0

Thus, the equation is satisfied when:

2x 0

or

1 - 6x 0

Solving these equations, we get:

x 0

x frac16

Step 2: Finding the Roots of f2x 0

Now, we need to find the roots of f2x 0. We know that f2x is a transformation of f(x). To find the roots of f2x 0, we need to solve:

f2x 22x - 12(2x)^2 0

This simplifies to:

22x - 48x^2 0

Factoring this equation, we get:

2x(1 - 24x) 0

Solving for x, we obtain:

2x 0

1 - 24x 0

Which gives:

x 0

x frac1{24}

Alternative Method Using g(x)

Alternatively, we can use the function g(x) f(2x). Given the roots of f(x) 0 are 1 and -2, we can substitute these into g(x) f(2x).

For f(1) 0, we have:

g(2*1/2) f2*1/2 f1 0

For f(-2) 0, we have:

g(-1) f2*(-1) f-2 0

Thus, the roots of g(x) f2x are:

x 1/2 and x -1

Conclusion

In conclusion, when the roots of f(x) 0 are given as 1 and -2, the roots of f2x 0 are x 1/2 and x -1. This relationship can be derived through direct substitution or by using the transformation function g(x) f(2x).

By understanding these concepts, you can effectively solve similar equations and deepen your knowledge of quadratic equations and transformations.