Question

If $2+i$ is a root of $x^2-4x+c=0$, where $c$ is a real number, then the value of $c$ is

• A. 2
• B. 3
• C. 4
• D. 5
• E. 0

Hint:

Shortcut Tip: You can also just substitute $x = 2+i$ directly into the equation! $(2+i)^2 - 4(2+i) + c = 0 \implies (3 + 4i) - (8 + 4i) + c = 0 \implies -5 + c = 0 \implies c = 5$.

Answer 1

The Correct Option is D

Concept:
The Complex Conjugate Root Theorem states that if a polynomial has real coefficients, then any complex roots must occur in conjugate pairs. Furthermore, for a quadratic equation $ax^2 + bx + c = 0$, the product of the roots is equal to $c/a$.

Step 1: Identify the nature of the coefficients.
The given quadratic equation is $x^2 - 4x + c = 0$. We are told that $c$ is a real number. Since $1$, $-4$, and $c$ are all real coefficients, the Complex Conjugate Root Theorem applies.

Step 2: Determine the second root.
We are given that one root is $r_1 = 2 + i$. Because complex roots must occur in conjugate pairs, the second root must be its conjugate: $$r_2 = 2 - i$$

Step 3: Use the relationship for the product of roots.
For the standard quadratic equation $ax^2 + bx + c = 0$, the product of the roots $(r_1 \cdot r_2)$ equals $\frac{c}{a}$. In our equation, $a = 1$. Therefore: $$r_1 \cdot r_2 = \frac{c}{1} = c$$

Step 4: Calculate the product of the complex roots.
Multiply the two conjugate roots together using the difference of squares pattern $(A+B)(A-B) = A^2 - B^2$: $$c = (2 + i)(2 - i)$$ $$c = 2^2 - (i)^2$$

Step 5: Evaluate the final value of c.
Remember that the imaginary unit squared is negative one ($i^2 = -1$). $$c = 4 - (-1)$$ $$c = 4 + 1$$ $$c = 5$$ Hence the correct answer is (D) 5.