For the quadratic equation x² - x = 12, which of the following statements is correct? Statement #1: The equation has one solution. Statement #2: The equation has two solutions. Statement #3: The equation has no solutions. (1 point) Responses Statement #1 Statement #1 Statement #2 Statement #2 Statement #3

Answer 1

To determine how many solutions the quadratic equation \(x^2 - x = 12\) has, we first rearrange it into standard form:

\[ x^2 - x - 12 = 0 \]

Next, we can analyze the quadratic equation \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = -1\), and \(c = -12\). To find the number of solutions, we can calculate the discriminant \(D\) using the formula:

\[ D = b^2 - 4ac \]

Substituting the values of \(a\), \(b\), and \(c\):

\[ D = (-1)^2 - 4(1)(-12) = 1 + 48 = 49 \]

Since the discriminant \(D\) is positive (\(49\)), it indicates that the quadratic equation has two distinct real solutions.

Therefore, the correct statement is:

Statement #2: The equation has two solutions.