The quadratic equation 4x^2 - 12x + 9 = 0 can be factored as: (2023)

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Question: The quadratic equation 4x^2 - 12x + 9 = 0 can be factored as: (2023)

Options:

Correct Answer: (2x - 3)(2x - 3)

Exam Year: 2023

Solution:

The equation can be factored as (2x - 3)(2x - 3) = 0, indicating a perfect square.

The quadratic equation 4x^2 - 12x + 9 = 0 can be factored as: (2023)

The quadratic equation 4x^2 - 12x + 9 = 0 can be factored as: (2023)

Step 1: Start with the quadratic equation: 4x^2 - 12x + 9 = 0.
Step 2: Identify the coefficients: a = 4, b = -12, c = 9.
Step 3: Check if the equation can be factored by looking for two numbers that multiply to (a * c) = 4 * 9 = 36 and add to b = -12.
Step 4: The numbers -6 and -6 work because -6 * -6 = 36 and -6 + -6 = -12.
Step 5: Rewrite the middle term (-12x) using -6 and -6: 4x^2 - 6x - 6x + 9 = 0.
Step 6: Group the terms: (4x^2 - 6x) + (-6x + 9) = 0.
Step 7: Factor out the common factors in each group: 2x(2x - 3) - 3(2x - 3) = 0.
Step 8: Notice that (2x - 3) is common in both groups: (2x - 3)(2x - 3) = 0.
Step 9: This can be written as (2x - 3)^2 = 0, indicating a perfect square.

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