Understanding the Problem: Finding the Value of 'a'

We are given a quadratic equation, \(x^2 - ax + 1 = 0\), and we are told that one of its roots is the complex number \(2 + i\). Our goal is to find the value of 'a'. This is a common type of problem involving roots of polynomial equations.

Using the Complex Conjugate Root Theorem

The given equation \(x^2 - ax + 1 = 0\) has coefficients 1, -a, and 1. If we assume that 'a' is a real number, then all the coefficients of this quadratic equation are real. A fundamental theorem in algebra states that if a polynomial equation with real coefficients has a complex number \(z\) as a root, then its complex conjugate \(\bar{z}\) must also be a root.

In this case, one given root is \(x_1 = 2 + i\). Since the coefficients of the equation are real (assuming 'a' is real), the complex conjugate of \(2 + i\), which is \(2 - i\), must also be a root. Let's call the second root \(x_2 = 2 - i\). This theorem is crucial when dealing with complex roots of equations like this quadratic equation.

Applying the Sum of Roots Property

For a general quadratic equation in the form \(Ax^2 + Bx + C = 0\), the sum of its roots is given by the formula \(\text{Sum of roots} = -\frac{B}{A}\). The product of its roots is given by \(\text{Product of roots} = \frac{C}{A}\).

Our equation is \(x^2 - ax + 1 = 0\). Comparing this to the general form, we have:

• \(A = 1\) (coefficient of \(x^2\))
• \(B = -a\) (coefficient of \(x\))
• \(C = 1\) (constant term)

Using the sum of roots formula for this specific equation:

\(\text{Sum of roots} = -\frac{-a}{1} = a\)

We know the two roots are \(x_1 = 2 + i\) and \(x_2 = 2 - i\). Let's calculate their sum:

\(x_1 + x_2 = (2 + i) + (2 - i)\)

\(x_1 + x_2 = 2 + i + 2 - i\)

\(x_1 + x_2 = (2 + 2) + (i - i)\)

\(x_1 + x_2 = 4 + 0i\)

\(x_1 + x_2 = 4\)

Determining the Value of 'a'

We found that the sum of the roots from the formula is equal to 'a', and the calculated sum of the specific roots \(2+i\) and \(2-i\) is 4. Therefore, we can equate these two expressions for the sum of roots:

\(a = 4\)

Thus, the value of a is 4.

This shows how the property of complex conjugate roots and the sum of roots formula help in finding the value of 'a' in the given quadratic equation when one root is known.