Question

(x-a)(x-b)=x²+x+4
1/a²+1/b²

Answer 1

$\frac{ - 7}{16}$

Step-by-step explanation:

$Given, \: \: (x-a)(x-b)=x²+x+4 \\ {x}^{2} - ax - bx + ab = {x}^{2} + x + 4 \\ - ax - bx + ab = x + 4 \\ - (a + b)x + ab = 1 \times x + 4 \\By \: \: comparison, \\ - (a + b) = 1 \: \: \: and \: \: \: ab = 4 \\ = > \: a + b = - 1 \: \: \: and \: \: \: ab = 4 \\ Now, \: \: \frac{1}{ {a}^{2} } + \frac{1}{ {b}^{2} } \\ = \frac{ {b}^{2} + {a}^{2} }{ {a}^{2} {b}^{2} } \\ = \frac{ {a}^{2} + {b}^{2} }{(ab)^{2} } \\ = \frac{(a + b)^{2} - 2ab}{(ab)^{2} } \\ putting \: \: values\\ = \frac{( - 1)^{2} - 2 \times 4}{(4)^{2} } \\ = \frac{1 - 8}{16} \\ = \frac{ - 7}{16}$