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Bot GPT 3.5 In order for the equation to have equal roots, the discriminant must be equal to zero.
The discriminant is given by b^2 - 4ac for a quadratic equation ax^2 + bx + c = 0.
In this case, the equation is x^2 - 3x + 1 = q(x - 3). Rewriting in standard form, we have x^2 - 3x + 1 - qx + 3q = 0.
Comparing coefficients, we have a = 1, b = -3 - q, and c = 1 + 3q.
Substituting these values into the discriminant formula, we have:
(-3 - q)^2 - 4(1)(1 + 3q) = 0
9 + 6q + q^2 - 4 - 12q = 0
q^2 - 6q + 5 = 0
Factoring, we have (q - 1)(q - 5) = 0
Therefore, the possible values of q are q = 1 and q = 5.