Introduction

Understanding the conditions under which a quadratic equation can have four real roots involves examining the factors of the equation and the discriminant of the quadratics derived from it. This article will delve into the process of determining the value of k for which the equation x^2 - 4x 3 k - 1 will have four real roots. We will break down the problem step-by-step to provide a clear and detailed explanation.

Step 1: Analyze the Quadratic Equation

The given quadratic equation is x^2 - 4x 3 k - 1. We can rewrite this equation by rearranging terms:

x^2 - 4x 4 - 4 3 k - 1

This simplifies to:

x^2 - 4x 4 - k - 2 0

or equivalently,

(x - 2)^2 - k - 2 0

This equation can be split into two cases based on the definition of the absolute value of x:

Case 1: ( x geq 0 )

In this case, we have:

(x - 2)^2 k 2

To find real roots, the right-hand side must be non-negative:

k 2 geq 0

This is always true for any real value of k.

Case 2: ( x

In this case, we have:

(-x - 2)^2 k 2

or, equivalently,

(x 2)^2 k 2

Similarly, for real roots, the right-hand side must be non-negative:

k 2 geq 0

Again, this is always true for any real value of k.

Step 2: Condition for Four Real Roots

To have four real roots overall, both quadratics must have two distinct real roots. This requires that their discriminants are positive. However, in our case, we are already at a point where both discriminants are always non-negative, as shown in the previous step. Therefore, we need to consider the product of the roots:

Product of Roots

From the equation (x - 2)^2 - k - 2 0, the roots of the quadratics are:

x 2 pm sqrt{k 2}

The product of these roots is:

(2 sqrt{k 2}) (2 - sqrt{k 2}) 4 - (k 2) 2 - k

The product of the roots of the other quadratic is equivalent. For the equation to have four real roots, the product of the roots must be non-zero:

2 - k eq 0

This implies:

k eq 2

Step 3: Conclusion

Thus, the value of k for which the equation x^2 - 4x 3 k - 1 will have four real roots is:

k eq 2

Therefore, the equation will have four real roots if and only if k eq 2.

Summary and Conclusion

Combining all the steps, we can conclude that the value of k for which the provided equation has four real roots is any value except k 2. This involves examining the conditions for the discriminants of the quadratics and ensuring that the product of the roots is non-zero.