Introduction

Understanding the conditions for a quadratic equation to have real roots is crucial for solving various mathematical and practical problems. In this article, we will explore the value of the variable a for which the quadratic equation x2 ax 4 0 has real roots. We will use the concept of the discriminant to determine these values, ensuring a comprehensive understanding of the topic.

The Role of the Discriminant

The discriminant Delta; is a key factor in determining the nature of the roots of a quadratic equation. The general form of a quadratic equation is x2 bx c 0, and the discriminant is given by:

Delta; b2 - 4ac

For the equation x2 ax 4 0, we can set b a and c 4. Thus, the discriminant becomes:

Delta; a2 - 4 × 4 a2 - 16

Conditions for Real Roots

To have real roots, the discriminant must be non-negative. Therefore, we need to solve:

a2 - 16 ge; 0

This inequality can be simplified as follows:

(a 4)(a - 4) ge; 0

The solution to this inequality is:

a le; -4 or a ge; 4

These are the values of a for which the given quadratic equation will have real roots.

Summary and Conclusion

In conclusion, the values of the variable a for which the quadratic equation x2 ax 4 0 has real roots are a le; -4 or a ge; 4. This is determined using the discriminant, which must be greater than or equal to zero for real roots to exist. Understanding these conditions is fundamental in solving quadratic equations and fundamental in many areas of mathematics and physics.