If the roots of the equation x^2 - 6x + k = 0 are 3 and 3, what is the value of

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Question: If the roots of the equation x^2 - 6x + k = 0 are 3 and 3, what is the value of k? (2020)

Options:

Correct Answer: 9

Exam Year: 2020

Solution:

The equation can be expressed as (x - 3)^2 = 0, which expands to x^2 - 6x + 9 = 0. Thus, k = 9.

If the roots of the equation x^2 - 6x + k = 0 are 3 and 3, what is the value of k? (2020)

If the roots of the equation x^2 - 6x + k = 0 are 3 and 3, what is the value of k? (2020)

Step 1: Identify the given equation, which is x^2 - 6x + k = 0.
Step 2: Note that the roots of the equation are both 3. This means the equation can be written in the form (x - 3)(x - 3) = 0.
Step 3: Rewrite (x - 3)(x - 3) as (x - 3)^2 = 0.
Step 4: Expand (x - 3)^2 using the formula (a - b)^2 = a^2 - 2ab + b^2.
Step 5: Calculate (x - 3)^2 = x^2 - 2(3)x + 3^2 = x^2 - 6x + 9.
Step 6: Compare the expanded equation x^2 - 6x + 9 with the original equation x^2 - 6x + k.
Step 7: Since both equations must be equal, set k equal to 9.

Quadratic Equations – Understanding the standard form of quadratic equations and how to find roots.
Factoring – Recognizing that a perfect square trinomial can be factored into a binomial squared.
Roots of Equations – Identifying the relationship between the coefficients of a quadratic equation and its roots.