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Let u and v be the solutions to 3x^2 + 5x + 7 = -4x^2 + 3x - 14. Find 1/u^2 + 1/v^2.

 Apr 2, 2024
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We can solve the equation for u and v first:

Combine like terms: 7x^2 + 2x - 21 = 0

Factor the equation: (x - 3)(7x + 7) = 0

Therefore, the solutions are x = 3 and x = -1. These correspond to u = 3 and v = -1.

Now, we need to find 1/u^2 + 1/v^2:

Substitute u = 3 and v = -1: 1/3^2 + 1/(-1)^2

Simplify: (1/9) + (1) = 1/9 + 9/9

Combine terms: 10/9

Therefore, 1/u^2 + 1/v^2 = 10/9.

 Apr 2, 2024

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