Determining the Value of k for No Real Roots in a Quadratic Equation

The quadratic equation x2 - 5kx 16 0 can reveal interesting patterns and solutions, particularly when considering the conditions under which it has no real roots. In this article, we will explore how the discriminant (D) can be used to determine the value of k that would make the roots of the equation non-real. This involves a step-by-step analysis and understanding of the quadratic formula and the discriminant's role in solving such equations.

Understanding the Discriminant

The discriminant of a quadratic equation ax2 bx c 0 is given by D b2 - 4ac. The discriminant provides crucial information about the roots of the equation:

If D 0, the equation has two distinct real roots. If D 0, the equation has exactly one real root (a repeated root). If D 0, the equation has no real roots (the roots are complex).

Given the equation x2 - 5kx 16 0, we can identify the coefficients as follows: a 1, b -5k, and c 16.

Setting Conditions for Non-Real Roots

To ensure that the quadratic equation has no real roots, we must set the discriminant less than zero (D 0):

[D b^2 - 4ac 0]

Substituting the values of a, b, and c, we get:

[(-5k)^2 - 4(1)(16) 0]

Simplifying this expression, we get:

[25k^2 - 64 0]

This can be further simplified to:

[25k^2 64]

Dividing both sides by 25, we obtain:

[k^2 frac{64}{25}]

Hence, the values of k that satisfy this inequality are:

[ -frac{8}{5} k frac{8}{5}]

Conclusion and Summary

In conclusion, the value of k that ensures the quadratic equation x2 - 5kx 16 0 has no real roots must lie within the interval ( -frac{8}{5} k frac{8}{5} ). This is achieved by setting the discriminant to be strictly less than zero.

Understanding and applying the concept of the discriminant not only helps in solving quadratic equations but also deepens the insights into the nature of their roots. If you have any more questions or need further assistance, feel free to explore related resources and examples.