Final answer:
To find the vertical asymptotes of the function f(x) = (3x² - 14x + 16) / (2x² - 10x + 12), we need to determine the values of x where the denominator of the function is equal to zero. The vertical asymptotes are x = 3 and x = 2.
Step-by-step explanation:
To find the vertical asymptotes of the function f(x) = (3x² - 14x + 16) / (2x² - 10x + 12), we need to determine the values of x where the denominator of the function is equal to zero. The denominator is a quadratic equation of the form ax² + bx + c = 0, where a = 2, b = -10, and c = 12. We can use the quadratic formula to solve for x:
x = (-b ± √(b² - 4ac)) / (2a)
Substituting the values of a, b, and c into the quadratic formula, we get:
x = (-(-10) ± √((-10)² - 4(2)(12))) / (2(2))
x = (10 ± √(100 - 96)) / 4
x = (10 ± √4) / 4
x = (10 ± 2) / 4
Therefore, the values of x are:
x = (10 + 2) / 4 = 12 / 4 = 3
x = (10 - 2) / 4 = 8 / 4 = 2
So, the vertical asymptotes of the function f(x) = (3x² - 14x + 16) / (2x² - 10x + 12) are x = 3 and x = 2.