Solve for x:
x^2+1/x^2==171
Bring x^2+1/x^2 together using the common denominator x^2:
x^4+1/x^2==171
Multiply both sides by x^2:
x^4+1==171 x^2
Subtract 171 x^2 from both sides:
x^4-171 x^2+1==0
Substitute y==x^2:
y^2-171 y+1==0
Subtract 1 from both sides:
y^2-171 y==-1
Add 29241/4 to both sides:
y^2-171 y+29241/4==29237/4
Write the left hand side as a square:
(y-171/2)^2==29237/4
Take the square root of both sides:
y-171/2==(13 Sqrt[173])/2 or y-171/2==-((13 Sqrt[173])/2)
Add 171/2 to both sides:
y==171/2+(13 Sqrt[173])/2 or y-171/2==-((13 Sqrt[173])/2)
Substitute back for y==x^2:
x^2==171/2+(13 Sqrt[173])/2 or y-171/2==-((13 Sqrt[173])/2)
Take the square root of both sides:
x==Sqrt[171/2+(13 Sqrt[173])/2] or x==-Sqrt[171/2+(13 Sqrt[173])/2] or y-171/2==-((13 Sqrt[173])/2)
171/2+(13 Sqrt[173])/2 == 169/4+(13 Sqrt[173])/2+173/4 == 169+26 Sqrt[173]+173/4 == 169+26 Sqrt[173]+(Sqrt[173])^2/4 == (Sqrt[173]+13)^2/4:
x==1/2 (Sqrt[173]+13) or x==-Sqrt[171/2+(13 Sqrt[173])/2] or y-171/2==-((13 Sqrt[173])/2)
171/2+(13 Sqrt[173])/2 == 169/4+(13 Sqrt[173])/2+173/4 == 169+26 Sqrt[173]+173/4 == 169+26 Sqrt[173]+(Sqrt[173])^2/4 == (Sqrt[173]+13)^2/4:
x==1/2 (13+Sqrt[173]) or x==-1/2 (Sqrt[173]+13) or y-171/2==-((13 Sqrt[173])/2)
Add 171/2 to both sides:
x==1/2 (13+Sqrt[173]) or x==1/2 (-13-Sqrt[173]) or y==171/2-(13 Sqrt[173])/2
Substitute back for y==x^2:
x==1/2 (13+Sqrt[173]) or x==1/2 (-13-Sqrt[173]) or x^2==171/2-(13 Sqrt[173])/2
Take the square root of both sides:
x==1/2 (13+Sqrt[173]) or x==1/2 (-13-Sqrt[173]) or x==Sqrt[171/2-(13 Sqrt[173])/2] or x==-Sqrt[171/2-(13 Sqrt[173])/2]
171/2-(13 Sqrt[173])/2 == 169/4-(13 Sqrt[173])/2+173/4 == 169-26 Sqrt[173]+173/4 == 169-26 Sqrt[173]+(Sqrt[173])^2/4 == (Sqrt[173]-13)^2/4:
x==1/2 (13+Sqrt[173]) or x==1/2 (-13-Sqrt[173]) or x==1/2 (Sqrt[173]-13) or x==-Sqrt[171/2-(13 Sqrt[173])/2]
171/2-(13 Sqrt[173])/2 == 169/4-(13 Sqrt[173])/2+173/4 == 169-26 Sqrt[173]+173/4 == 169-26 Sqrt[173]+(Sqrt[173])^2/4 == (Sqrt[173]-13)^2/4:
x==1/2 (13+Sqrt[173]) or x==1/2 (-13-Sqrt[173]) or x==1/2 (Sqrt[173]-13) or x==-1/2 (Sqrt[173]-13)
[1/2 (13+Sqrt[173])] - 1 / [1/2 (13+Sqrt[173])] = 13 or [1/2 (Sqrt[173]-13)] - 1 / [1/2 (Sqrt[173]-13)] = -13
+26367 If
\(x^2 + \dfrac{1}{x^2} = 171\),
then what are the possible values of
\(x - \dfrac{1}{x}\)?
\(\begin{array}{|rcll|} \hline \left(x - \dfrac{1}{x}\right)^2 &=& x^2-2*x*\dfrac{1}{x}+\dfrac{1}{x^2} \\\\ &=& x^2+\dfrac{1}{x^2}-\dfrac{2x}{x} \\\\ &=& x^2+\dfrac{1}{x^2}-2 \quad | \quad \mathbf{x^2 + \dfrac{1}{x^2} = 171} \\\\ &=& 171-2 \\\\ &=& \mathbf{169} \\ \hline x - \dfrac{1}{x} &=& \pm \sqrt{169} \\\\ \mathbf{x - \dfrac{1}{x}} &=& \mathbf{\pm 13} \\ \hline \end{array}\)
