Question

Which of the following describes the root(s) of the equation ( 9x² = 6x - 1 )?

(A) Exactly one real root
(B) Two distinct imaginary roots
(C) Exactly one imaginary root
(D) Two distinct real roots

Answer 1

The equation 9x² = 6x - 1 leads to a discriminant of zero when rearranged and evaluated, indicating exactly one real root. Thus, the correct answer is (A) Exactly one real root.

Step-by-step explanation:

The equation given is 9x² = 6x - 1. To find out the nature of the roots, we should convert this equation into the standard quadratic form ax² + bx + c = 0. Doing so involves moving the terms on the right side to the left side to give us 9x² - 6x + 1 = 0.

Now, we apply the quadratic formula, x = √{-b ± √{b²-4ac}}/2a, to determine the roots. For our equation, a = 9, b = -6, and c = 1. We then compute the discriminant, which is b² - 4ac, and find that in our case it is 36 - 4(9)(1) = 36 - 36 = 0. The discriminant being zero indicates that there is exactly one real root, since the square root of zero is zero, which results in one value for x when substituted into the quadratic formula.

Therefore, the correct answer corresponding to the roots of the equation 9x² = 6x - 1 is (A) Exactly one real root.