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swer gete pand x² + ₂ 332474 w the roots of 2x² + 3x - 1 = 0 are t and b, find the values of (a) t² + b² (b) (t - b)² 2 + 1/b2)(b2+1/t2) ​

User Jbrown
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1 Answer

4 votes

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.

User Jitendra Ahuja
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