To find the x-intercepts of the function \( y = x^2 - x - 120 \), we need to set \( y \) to 0 and solve for \( x \):
\[ 0 = x^2 - x - 120 \]
This is a quadratic equation. We can solve it using the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation, \( a = 1 \), \( b = -1 \), and \( c = -120 \). Plugging in these values:
- Calculate the discriminant:
\[ b^2 - 4ac = (-1)^2 - 4(1)(-120) = 1 + 480 = 481 \]
- Now substitute into the quadratic formula:
\[ x = \frac{-(-1) \pm \sqrt{481}}{2(1)} = \frac{1 \pm \sqrt{481}}{2} \]
Calculating \( \sqrt{481} \): \[ \sqrt{481} \approx 21.93 \]
Now calculate the two possible values for \( x \):
- \( x_1 = \frac{1 + 21.93}{2} \approx \frac{22.93}{2} \approx 11.465 \)
- \( x_2 = \frac{1 - 21.93}{2} \approx \frac{-20.93}{2} \approx -10.465 \)
So, the x-intercepts are approximately \( (11.465, 0) \) and \( (-10.465, 0) \).
Now let's match these values with the options provided:
- (–120, 0) - Not an x-intercept
- (–11.466, 0) - Not correct, it should be approximately –10.465
- (–10.466, 0) - Close enough to –10.465 (acceptable error)
- (0, –120) - Not an x-intercept
- (10.466, 0) - Close enough to 11.465 (acceptable error)
- (11.466, 0) - Close enough to 11.465 (acceptable error)
The correct x-intercepts, checking against the options, are:
- (–10.466, 0)
- (10.466, 0)
- (11.466, 0)
Thus, the correct answers are (–10.466, 0), (10.466, 0), and (11.466, 0).