Introduction

The equation (x^4 - 2x^2 - 3 0) is a quartic equation, which may appear complicated at first glance. However, by employing a strategic substitution and leveraging the properties of quadratic equations, we can determine whether this equation has any real roots. This article will delve into the methods and provide a detailed analysis of the equation's roots.

Substitution Method

To simplify the given equation, we can make the substitution (y x^2). This transforms the quartic equation into a quadratic equation, which is much easier to handle. The transformed equation is:

(y^2 - 2y - 3 0)

Quadratic Equation Analysis

Now, we solve the quadratic equation using the quadratic formula:

[ y frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

Here, the coefficients are (a 1), (b -2), and (c -3). Plugging these values into the formula, we get:

[ y frac{2 pm sqrt{(-2)^2 - 4 cdot 1 cdot (-3)}}{2 cdot 1} frac{2 pm sqrt{4 12}}{2} frac{2 pm sqrt{16}}{2} ]

Simplifying this, we find:

[ y frac{2 pm 4}{2} ]

This yields two possible values for (y):

[ y frac{6}{2} 3 ] [ y frac{-2}{2} -1 ]

We can see that (y) cannot be negative because (y x^2) and the square of a real number cannot be negative. Therefore, the only valid solution is (y 3).

Elimination of Real Solutions

Since (y x^2) and (x^2) must be non-negative, we have:

[ x^2 3 Rightarrow x pm sqrt{3} ]

However, this only provides the potential real roots. We must substitute back into the original equation to verify:

[ (sqrt{3})^4 - 2(sqrt{3})^2 - 3 9 - 6 - 3 0 ]

And similarly for [ x -sqrt{3} ]:

[ (-sqrt{3})^4 - 2(-sqrt{3})^2 - 3 9 - 6 - 3 0 ]

Therefore, the only real solutions are (x pm sqrt{3}).

Further Analysis with Derivatives

To confirm the nature of these roots, we can analyze the first and second derivatives of the function (f(x) x^4 - 2x^2 - 3) to determine the critical points and the nature of the function around these points.

First derivative:

[ f'(x) 4x^3 - 4x 4x(x^2 - 1) ]

Solving for critical points:

[ 4x(x^2 - 1) 0 Rightarrow x 0, x pm 1 ]

Second derivative:

[ f''(x) 12x^2 - 4 ]

Evaluating the second derivative at the critical points:

At (x 0), [ f''(0) -4 At (x 1), [ f''(1) 8 ] At (x -1), [ f''(-1) 8 ]

This indicates that (x 0) is a local minimum, and (x pm 1) are local maxima. Evaluating the function at these points:

[ f(0) -3 ]

[ f(1) f(-1) -2 ]

Since the minimum value of the function is -3, the function (f(x)) is always greater than -3, thus proving that there are no real roots for the equation (x^4 - 2x^2 - 3 0).

Conclusion

Through the substitution method and the analysis of derivatives, we have shown that the equation (x^4 - 2x^2 - 3 0) does not have any real roots. The roots that we found are complex roots, which are (x pm i sqrt{2}).

The final answer is: There are no real roots.