Final answer:
The quadratic equation with roots √3 + 1/2 and √3 - 1/2 is x^2 - 2√3x + 11/4 = 0, derived from the sum and product of the roots as per Vièta's formulas.
Step-by-step explanation:
To write the quadratic equation with the given roots, firstly, we need to understand that according to Vièta's formulas, the sum and the product of the roots of the quadratic equation ax^2+bx+c=0 can be found using the relations -b/a for the sum and c/a for the product of the roots. In our case, the roots are √3 + 1/2 and √3 - 1/2.
To find the sum of the roots, we add them together:
S = (√3 + 1/2) + (√3 - 1/2) = 2√3
To find the product of the roots, we multiply them:
P = (√3 + 1/2) * (√3 - 1/2) = 3 - 1/4 = 11/4
Since the coefficient with x^2 is 1 (implied by 'a' being equal to 1), our quadratic equation can be constructed as follows:
x^2 - (sum of roots)x + (product of roots) = 0
x^2 - 2√3x + 11/4 = 0
This is the required quadratic equation. To further validate, one could apply the quadratic formula to this equation and retrieve the original roots as solutions.