Introduction

In this article, we will walk through the process of finding a quadratic equation whose roots are given as 7±2√5. This is a crucial concept in algebra and forms the foundation for many advanced topics in mathematics. Understanding this process will not only help in solving specific problems but also enhance your problem-solving skills.

Understanding the Roots and the Quadratic Equation

The roots of a quadratic equation are the values of the variable (typically x) that satisfy the equation. If r1 and r2 are the roots of a quadratic equation, the equation can be expressed in the form:

x2 - (r1 r2)x (r1 · r2) 0

Step-by-Step Solution

Step 1: Identify the Roots

The given roots are:

r1 7 2√5, r2 7 - 2√5

Step 2: Calculate the Sum of the Roots

The sum of the roots, r1 r2, is calculated as follows:

r1 r2 (7 2√5) (7 - 2√5) 14

Step 3: Calculate the Product of the Roots

The product of the roots, r1 · r2, is calculated as follows:

r1 · r2 (7 2√5)(7 - 2√5) 72 - (2√5)2 49 - 20 29

Step 4: Write the Quadratic Equation

Substitute the sum and product of the roots into the quadratic equation form:

x2 - (r1 r2)x (r1 · r2) 0

Therefore, the quadratic equation is:

x2 - 14x 29 0

Conclusion

The quadratic equation whose roots are 7±2√5 is:

boxed{x2 - 14x 29 0}

Additional Notes

Understanding the process of finding a quadratic equation from its roots is not only useful for solving problems but also helps in grasping the structure and properties of quadratic equations. This method can be applied to similar problems with different roots. If you encounter any further questions or need further assistance, feel free to reach out.