Question

What is the nature of the roots of the quadratic equation X$^2$--6X+9=0

• A. Real and distinct
• B. Real and equal
• C. Complex
• D. Imaginary

Hint:

Whenever a quadratic equation is a perfect square trinomial (like \(X^2 - 2kX + k^2 = (X - k)^2 = 0\)), its discriminant is always zero, and its roots are always real and equal.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The nature of the roots of a standard quadratic equation of the form \(aX^2 + bX + c = 0\) is determined by its discriminant (\(D\)).
The discriminant reveals whether the roots are real, distinct, equal, or complex.
Key Formula or Approach:
The formula for the discriminant (\(D\)) of a quadratic equation is:
\[ D = b^2 - 4ac \] The rules for the nature of the roots are:
- If \(D > 0\), the roots are real and distinct.
- If \(D = 0\), the roots are real and equal.
- If \(D < 0\), the roots are complex or imaginary.

Step 2: Detailed Explanation:
Given the quadratic equation:
\[ X^2 - 6X + 9 = 0 \] Comparing this with the standard form \(aX^2 + bX + c = 0\), we identify the coefficients as:
- \(a = 1\)
- \(b = -6\)
- \(c = 9\)
Now, calculate the discriminant (\(D\)):
\[ D = (-6)^2 - 4(1)(9) \] \[ D = 36 - 36 \] \[ D = 0 \] Since the discriminant is exactly equal to zero (\(D = 0\)), the quadratic equation has two real and identical roots.
We can also verify this by factoring the quadratic expression directly:
\[ X^2 - 6X + 9 = (X - 3)^2 = 0 \] This yields the roots:
\[ X_1 = 3, \quad X_2 = 3 \] Thus, the roots are indeed real and equal.

Step 3: Final Answer:
The nature of the roots of the given quadratic equation is real and equal.