Final answer:
To find the values of k for which the quadratic equation f(x) = 18kx² - 3kx - 1/4 has two real roots, we need to find the discriminant of the equation. By setting the discriminant greater than 0 and solving for k, we find that k can be 0 or 1/9.
Step-by-step explanation:
To determine the values of k for which the quadratic equation f(x) = 18kx² - 3kx - 1/4 has two real roots, we need to find the discriminant of the equation. The discriminant formula is b² - 4ac. In this equation, a = 18k, b = -3k, and c = -1/4. If the discriminant is greater than 0, the equation will have two distinct real roots. So we set the discriminant greater than 0 and solve for k:
(-3k)² - 4(18k)(-1/4) > 0
-9k² + 18k/4 > 0
-9k² + 4.5k > 0
This is a quadratic inequality. To solve this inequality, we can factor or use the quadratic formula. Since the quadratic equation is not easily factored, we can use the quadratic formula to find the roots of the equation:
k = (-4.5 ± sqrt((4.5)² - 4(-9)(0)))/(2(-9))
k = (-4.5 ± sqrt(20.25 + 0))/(2(-9))
k = (-4.5 ± sqrt(20.25))/(2(-9))
k = (-4.5 ± 4.5)/(2(-9))
k = 0 or k = 1/9