Answer:
Explanation:
To find the values of the given expressions, we first need to find the values of t and b, which are the roots of the equation 2x² + 3x - 1 = 0.
Using the quadratic formula, we can solve for x:
x = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = 2, b = 3, and c = -1. Plugging in these values, we can calculate t and b:
t = (-3 + √(3² - 4(2)(-1))) / (2(2))
t = (-3 + √(9 + 8)) / 4
t = (-3 + √17) / 4
b = (-3 - √(3² - 4(2)(-1))) / (2(2))
b = (-3 - √(9 + 8)) / 4
b = (-3 - √17) / 4
Now, let's calculate the values of the given expressions:
(a) t² + b²
Substituting the values of t and b:
(t² + b²) = ((-3 + √17) / 4)² + ((-3 - √17) / 4)²
= (9 - 6√17 + 17) / 16 + (9 + 6√17 + 17) / 16
= (35 + 12√17) / 16 + (35 - 12√17) / 16
= 70 / 16
= 35 / 8
(b) (t - b)²
Substituting the values of t and b:
(t - b)² = ((-3 + √17) / 4 - (-3 - √17) / 4)²
= (2√17)²
= 4(17)
= 68
(c) (2 + 1/b²)(b² + 1/t²)
Substituting the values of t and b:
(2 + 1/b²)(b² + 1/t²) = (2 + 1/((-3 - √17) / 4)²)((-3 - √17) / 4)² + 1/((-3 + √17) / 4)²)
= (2 + 16 / (3 + √17)²)((-3 - √17) / 4)² + 16 / (3 - √17)²)
= (2 + 16 / (3 + √17)²)((-3 - √17) / 4)² + 16 / (3 - √17)²)
The final expression cannot be further simplified without knowing the exact values of √17.