the radical equation: \(\sqrt{{q^2\over 2}+11} = q - 1\)
i then have to square both sides; but how should i square q - 1?
and after i squared both sides, what should i do next?
Solve for q:
sqrt(q^2/2 + 11) = q - 1
Raise both sides to the power of two:
q^2/2 + 11 = (q - 1)^2
Expand out terms of the right hand side:
q^2/2 + 11 = q^2 - 2 q + 1
Subtract q^2 - 2 q + 1 from both sides:
-q^2/2 + 2 q + 10 = 0
Multiply both sides by -2:
q^2 - 4 q - 20 = 0
Add 20 to both sides:
q^2 - 4 q = 20
Add 4 to both sides:
q^2 - 4 q + 4 = 24
Write the left hand side as a square:
(q - 2)^2 = 24
Take the square root of both sides:
q - 2 = 2 sqrt(6) or q - 2 = -2 sqrt(6)
Add 2 to both sides:
q = 2 + 2 sqrt(6) or q - 2 = -2 sqrt(6)
Add 2 to both sides:
q = 2 + 2 sqrt(6) or q = 2 - 2 sqrt(6)
sqrt(q^2/2 + 11) ⇒ sqrt(1/2 (2 - 2 sqrt(6))^2 + 11) = sqrt(25 - 4 sqrt(6)) ≈ 3.89898
q - 1 ⇒ (2 - 2 sqrt(6)) - 1 = 1 - 2 sqrt(6) ≈ -3.89898:
So this solution is incorrect
sqrt(q^2/2 + 11) ⇒ sqrt(1/2 (2 + 2 sqrt(6))^2 + 11) = sqrt(25 + 4 sqrt(6)) ≈ 5.89898
q - 1 ⇒ (2 + 2 sqrt(6)) - 1 = 1 + 2 sqrt(6) ≈ 5.89898:
So this solution is correct
The solution is: q = 2 + 2 sqrt(6)
Solve for q:
sqrt(q^2/2 + 11) = q - 1
Raise both sides to the power of two:
q^2/2 + 11 = (q - 1)^2
Expand out terms of the right hand side:
q^2/2 + 11 = q^2 - 2 q + 1
Subtract q^2 - 2 q + 1 from both sides:
-q^2/2 + 2 q + 10 = 0
Multiply both sides by -2:
q^2 - 4 q - 20 = 0
Add 20 to both sides:
q^2 - 4 q = 20
Add 4 to both sides:
q^2 - 4 q + 4 = 24
Write the left hand side as a square:
(q - 2)^2 = 24
Take the square root of both sides:
q - 2 = 2 sqrt(6) or q - 2 = -2 sqrt(6)
Add 2 to both sides:
q = 2 + 2 sqrt(6) or q - 2 = -2 sqrt(6)
Add 2 to both sides:
q = 2 + 2 sqrt(6) or q = 2 - 2 sqrt(6)
sqrt(q^2/2 + 11) ⇒ sqrt(1/2 (2 - 2 sqrt(6))^2 + 11) = sqrt(25 - 4 sqrt(6)) ≈ 3.89898
q - 1 ⇒ (2 - 2 sqrt(6)) - 1 = 1 - 2 sqrt(6) ≈ -3.89898:
So this solution is incorrect
sqrt(q^2/2 + 11) ⇒ sqrt(1/2 (2 + 2 sqrt(6))^2 + 11) = sqrt(25 + 4 sqrt(6)) ≈ 5.89898
q - 1 ⇒ (2 + 2 sqrt(6)) - 1 = 1 + 2 sqrt(6) ≈ 5.89898:
So this solution is correct
The solution is: q = 2 + 2 sqrt(6)