Multiple choice

If $ \alpha , \beta $ are the roots of the equation $ x ^ { 2 } - 15 x + 1 = 0 $, then the value is $ \left( \frac { 1 } { \alpha } - 15 \right) ^ { - 2 } + \left( \frac { 1 } { \beta } - 15 \right) ^ { - 2 } $ is

  1. 225

  2. 900

  3. 223

  4. 0
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C Correct answer
Explanation

Given x^2 - 15x + 1 = 0, roots alpha and beta satisfy alpha + beta = 15 and alpha*beta = 1. The expression is (1/alpha - 15)^-2 + (1/beta - 15)^-2. Since 1/alpha - 15 = (1 - 15alpha)/alpha = -alpha^2/alpha = -alpha, the expression simplifies to (-alpha)^-2 + (-beta)^-2 = 1/alpha^2 + 1/beta^2 = (alpha^2 + beta^2)/(alpha*beta)^2. With alpha^2 + beta^2 = (alpha+beta)^2 - 2alpha*beta = 225 - 2 = 223, the result is 223/1 = 223.

AI explanation

Using the property of quadratic roots, we can replace alpha with 15 minus 1 over alpha to simplify the expression to 1 over beta minus 15 to the power of negative 2 plus 1 over alpha minus 15 to the power of negative 2. The value of 1 over alpha plus 1 over beta equals the sum of the roots, which is 15. When squaring the reciprocals, this becomes 1 over beta squared plus 1 over alpha squared, and we find the value of the squared sum is 223. The result is 223.

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