Question

15x^4-34x^3-22x^2+34x+15=0 and x-1/x=a find a

Answer 1

Final Answer : a = 2,4/15

Steps :
1) Given :

We know,
$x - \frac{1}{x} = a \: .then \: \\ = > {a}^{2} = {x}^{2} - 2 + { (\frac{1}{x} )}^{2} \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = {a}^{2} + 2 - - (1)$
2)
$15 {x}^{4} - 34 {x}^{3} - 22 {x}^{2} + 34x + 15 = 0 \\ divide \: by \: {x}^{2} \\ = > 15 {x}^{2} - 34x - 22 + \frac{34}{x} + \frac{15}{ {x}^{2} } = 0 \\ = > 15( {x}^{2} + \frac{1}{ {x}^{2} } ) - 34(x - \frac{1}{x} ) - 22 = 0 \\ = > 15( {a}^{2} + 2) - 34a - 22 = 0 \\ = > 15 {a}^{2} - 34a + 30 - 22 = 0 \\ = > 15 {a}^{2} - 34a + 8 = 0 \\ = > 15 {a}^{2} - 30a - 4a + 8 = 0 \\ = > 15a(a - 2) - 4(a - 2) = 0 \\ = > (a - 2)(15a - 4) = 0 \\ = > a = 2 \: \: or \: a = \frac{4}{15}$

3) If a is natural no. then, a = 2 ,