Question

if X equal 4 minus root 15 find the value of x plus one upon x and x minus one upon x​

Answer 1

Step-by-step explanation:

$Given: \\ x = 4 - \sqrt{15} \\ \therefore \: x + \frac{1}{x} = \frac{ {x}^{2} + 1 }{x} \\ = \frac{( {4 - \sqrt{15} )}^{2} + 1}{4 - \sqrt{15} } \\ = \frac{ {4}^{2} + (\sqrt{15})^{2} - 2 \times 4 \times \sqrt{15} + 1}{4 - \sqrt{15}} \: \\ = \frac{16 + 15 - 8 \sqrt{15} + 1 }{4 - \sqrt{15}} \\ = \frac{32 - 8 \sqrt{15} }{4 - \sqrt{15}} \\ = \frac{8(4 - \sqrt{15} )}{4 - \sqrt{15}} \\ = 8 \\ thus \: x + \frac{1}{x} = 8 \\ \\ now \: \\ x - \frac{1}{x} = \frac{ {x}^{2} - 1 }{x} \\ = \frac{( {4 - \sqrt{15} )}^{2} - 1}{4 - \sqrt{15} } \\ = \frac{ {4}^{2} + (\sqrt{15})^{2} - 2 \times 4 \times \sqrt{15} - 1}{4 - \sqrt{15}} \: \\ = \frac{16 + 15 - 8 \sqrt{15} - 1 }{4 - \sqrt{15}} \\ = \frac{30 - 8 \sqrt{15} }{4 - \sqrt{15}} \\ = \frac{2(15- 4\sqrt{15} )}{4 - \sqrt{15}} \\ = \frac{2 ( \sqrt{15} \times \sqrt{15} - 4\sqrt{15} )}{4 - \sqrt{15}} \\ = \frac{ - 2 \sqrt{15} ( - \sqrt{15} + 4 )}{4 - \sqrt{15}} \\ = \frac{ - 2 \sqrt{15} ( 4- \sqrt{15} )}{4 - \sqrt{15}} \\ = - 2 \sqrt{15} \\ thus \: \: \: \: x - \frac{1}{x} = - 2 \sqrt{15}$