Solve for x:
x^2+1/x^2 = 62
Bring x^2+1/x^2 together using the common denominator x^2:
(x^4+1)/x^2 = 62
Multiply both sides by x^2:
x^4+1 = 62 x^2
Subtract 62 x^2 from both sides:
x^4-62 x^2+1 = 0
Substitute y = x^2:
y^2-62 y+1 = 0
Subtract 1 from both sides:
y^2-62 y = -1
Add 961 to both sides:
y^2-62 y+961 = 960
Write the left hand side as a square:
(y-31)^2 = 960
Take the square root of both sides:
y-31 = 8 sqrt(15) or y-31 = -8 sqrt(15)
Add 31 to both sides:
y = 31+8 sqrt(15) or y-31 = -8 sqrt(15)
Substitute back for y = x^2:
x^2 = 31+8 sqrt(15) or y-31 = -8 sqrt(15)
Take the square root of both sides:
x = sqrt(31+8 sqrt(15)) or x = -sqrt(31+8 sqrt(15)) or y-31 = -8 sqrt(15)
31+8 sqrt(15) = 16+8 sqrt(15)+15 = 16+8 sqrt(15)+sqrt(15)^2 = (4+sqrt(15))^2:
x = 4+sqrt(15) or x = -sqrt(31+8 sqrt(15)) or y-31 = -8 sqrt(15)
31+8 sqrt(15) = 16+8 sqrt(15)+15 = 16+8 sqrt(15)+sqrt(15)^2 = (4+sqrt(15))^2:
x = 4+sqrt(15) or x = -4+sqrt(15) or y-31 = -8 sqrt(15)
Add 31 to both sides:
x = 4+sqrt(15) or x = -4-sqrt(15) or y = 31-8 sqrt(15)
Substitute back for y = x^2:
x = 4+sqrt(15) or x = -4-sqrt(15) or x^2 = 31-8 sqrt(15)
Take the square root of both sides:
x = 4+sqrt(15) or x = -4-sqrt(15) or x = sqrt(31-8 sqrt(15)) or x = -sqrt(31-8 sqrt(15))
31-8 sqrt(15) = 16-8 sqrt(15)+15 = 16-8 sqrt(15)+sqrt(15)^2 = (4-sqrt(15))^2:
x = 4+sqrt(15) or x = -4-sqrt(15) or x = 4-sqrt(15) or x = -sqrt(31-8 sqrt(15))
31-8 sqrt(15) = 16-8 sqrt(15)+15 = 16-8 sqrt(15)+sqrt(15)^2 = (4-sqrt(15))^2:
Answer: |x = 4+sqrt(15) or x = -4-sqrt(15) or x = 4-sqrt(15) or x = -4-sqrt(15)
Solve for x:
x^2+1/x^2 = 62
Bring x^2+1/x^2 together using the common denominator x^2:
(x^4+1)/x^2 = 62
Multiply both sides by x^2:
x^4+1 = 62 x^2
Subtract 62 x^2 from both sides:
x^4-62 x^2+1 = 0
Substitute y = x^2:
y^2-62 y+1 = 0
Subtract 1 from both sides:
y^2-62 y = -1
Add 961 to both sides:
y^2-62 y+961 = 960
Write the left hand side as a square:
(y-31)^2 = 960
Take the square root of both sides:
y-31 = 8 sqrt(15) or y-31 = -8 sqrt(15)
Add 31 to both sides:
y = 31+8 sqrt(15) or y-31 = -8 sqrt(15)
Substitute back for y = x^2:
x^2 = 31+8 sqrt(15) or y-31 = -8 sqrt(15)
Take the square root of both sides:
x = sqrt(31+8 sqrt(15)) or x = -sqrt(31+8 sqrt(15)) or y-31 = -8 sqrt(15)
31+8 sqrt(15) = 16+8 sqrt(15)+15 = 16+8 sqrt(15)+sqrt(15)^2 = (4+sqrt(15))^2:
x = 4+sqrt(15) or x = -sqrt(31+8 sqrt(15)) or y-31 = -8 sqrt(15)
31+8 sqrt(15) = 16+8 sqrt(15)+15 = 16+8 sqrt(15)+sqrt(15)^2 = (4+sqrt(15))^2:
x = 4+sqrt(15) or x = -4+sqrt(15) or y-31 = -8 sqrt(15)
Add 31 to both sides:
x = 4+sqrt(15) or x = -4-sqrt(15) or y = 31-8 sqrt(15)
Substitute back for y = x^2:
x = 4+sqrt(15) or x = -4-sqrt(15) or x^2 = 31-8 sqrt(15)
Take the square root of both sides:
x = 4+sqrt(15) or x = -4-sqrt(15) or x = sqrt(31-8 sqrt(15)) or x = -sqrt(31-8 sqrt(15))
31-8 sqrt(15) = 16-8 sqrt(15)+15 = 16-8 sqrt(15)+sqrt(15)^2 = (4-sqrt(15))^2:
x = 4+sqrt(15) or x = -4-sqrt(15) or x = 4-sqrt(15) or x = -sqrt(31-8 sqrt(15))
31-8 sqrt(15) = 16-8 sqrt(15)+15 = 16-8 sqrt(15)+sqrt(15)^2 = (4-sqrt(15))^2:
Answer: |x = 4+sqrt(15) or x = -4-sqrt(15) or x = 4-sqrt(15) or x = -4-sqrt(15)
