The equation x^2 - 6x + 9 = 0 has how many distinct roots?

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Question: The equation x^2 - 6x + 9 = 0 has how many distinct roots?

Options:

Correct Answer: 1

Solution:

The equation can be factored as (x - 3)(x - 3) = 0, indicating it has one distinct root (a double root).

The equation x^2 - 6x + 9 = 0 has how many distinct roots?

The equation x^2 - 6x + 9 = 0 has how many distinct roots?

Step 1: Identify the equation given, which is x^2 - 6x + 9 = 0.
Step 2: Recognize that this is a quadratic equation in the form of ax^2 + bx + c.
Step 3: Check if the equation can be factored. Look for two numbers that multiply to 9 (the constant term) and add to -6 (the coefficient of x).
Step 4: Notice that -3 and -3 multiply to 9 and add to -6.
Step 5: Write the factored form of the equation as (x - 3)(x - 3) = 0.
Step 6: Set each factor equal to zero: x - 3 = 0.
Step 7: Solve for x, which gives x = 3.
Step 8: Since both factors are the same, x = 3 is a double root, meaning it is the only root.
Step 9: Conclude that the equation has one distinct root.

Quadratic Equations – Understanding the nature of roots of quadratic equations, including distinct and repeated roots.
Factoring – Ability to factor quadratic expressions to find roots.
Discriminant – Using the discriminant (b^2 - 4ac) to determine the number of distinct roots.