Final answer:
The roots of the quadratic equation y = -3x^2 + 60x - 161 are approximately -6.143 and 2.476. The vertex of the quadratic equation is approximately (10.000, 139.000).
Step-by-step explanation:
To find the roots of the quadratic equation y = -3x^2 + 60x - 161, we can use the quadratic formula. The formula is x = (-b ± √(b^2 - 4ac)) / 2a. In this equation, a = -3, b = 60, and c = -161. Plugging in these values, we get x = (-60 ± √(60^2 - 4(-3)(-161))) / (2(-3)). Simplifying further, x = (-60 ± √(3600 - 1932)) / (-6). Continuing the calculation, x = (-60 ± √(1668)) / (-6). Finally, we can approximate the roots to three decimal places: x ≈ (-60 ± 40.854) / (-6). Therefore, the roots are approximately x ≈ -6.143 and x ≈ 2.476.
To find the vertex of the quadratic equation, we can use the formula x = -b / (2a). Plugging in the values for a and b from the equation, we get x = -60 / (2(-3)). Simplifying further, x = -60 / (-6). Thus, the x-coordinate of the vertex is x ≈ 10.000. To find the y-coordinate, we can substitute this value of x into the original equation y = -3x^2 + 60x - 161. Plugging in x ≈ 10.000, we get y = -3(10^2) + 60(10) - 161. Simplifying further, y = -3(100) + 600 - 161. Continuing the calculation, y = -300 + 600 - 161. Therefore, the y-coordinate of the vertex is y ≈ 139.000.