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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?

Guest Oct 11, 2017

#1**+1 **

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)**

Guest Oct 11, 2017

#1**+1 **

Best Answer

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)**

Guest Oct 11, 2017