# Solve for Q Q^(1/2)=Q^2 Q12=Q2
Subtract Q2 from both sides of the equation.
Q12-Q2=0
Find a common factor Q12 that is present in each term.
Q12-1(Q12)4
Substitute u for Q12.
(u)-1(u)4=0
Solve for u.
Factor the equation.
Remove parentheses.
u-1(u)4=0
Factor u out of u-1(u)4.
Raise u to the power of 1.
u-1(u)4=0
Factor u out of u1.
u⋅1-1(u)4=0
Factor u out of -1(u)4.
u⋅1+u(-1u3)=0
Factor u out of u⋅1+u(-1u3).
u(1-1u3)=0
u(1-1u3)=0
Rewrite 1 as 13.
u(13-1u3)=0
Since both terms are perfect cubes, factor using the difference of cubes formula, a3-b3=(a-b)(a2+ab+b2) where a=1 and b=u.
u((1-u)(12+1u+u2))=0
Factor.
Simplify.
One to any power is one.
u((1-u)(1+1u+u2))=0
Multiply u by 1.
u((1-u)(1+u+u2))=0
u((1-u)(1+u+u2))=0
Remove unnecessary parentheses.
u(1-u)(1+u+u2)=0
u(1-u)(1+u+u2)=0
u(1-u)(1+u+u2)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
u=0
1-u=0
1+u+u2=0
Set the first factor equal to 0.
u=0
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
1-u=0
Subtract 1 from both sides of the equation.
-u=-1
Multiply each term in -u=-1 by -1
Multiply each term in -u=-1 by -1.
(-u)⋅-1=(-1)⋅-1
Multiply (-u)⋅-1.
Multiply -1 by -1.
1u=(-1)⋅-1
Multiply u by 1.
u=(-1)⋅-1
u=(-1)⋅-1
Multiply -1 by -1.
u=1
u=1
u=1
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
1+u+u2=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=1, b=1, and c=1 into the quadratic formula and solve for u.
-1±12-4⋅(1⋅1)2⋅1
Simplify.
Simplify the numerator.
One to any power is one.
u=-1±1-4⋅(1⋅1)2⋅1
Multiply 1 by 1.
u=-1±1-4⋅12⋅1
Multiply -4 by 1.
u=-1±1-42⋅1
Subtract 4 from 1.
u=-1±-32⋅1
Rewrite -3 as -1(3).
u=-1±-1⋅32⋅1
Rewrite -1(3) as -1⋅3.
u=-1±-1⋅32⋅1
Rewrite -1 as i.
u=-1±i32⋅1
u=-1±i32⋅1
Multiply 2 by 1.
u=-1±i32
u=-1±i32
The final answer is the combination of both solutions.
u=-1-i32,-1+i32
u=-1-i32,-1+i32
The final solution is all the values that make u(1-u)(1+u+u2)=0 true.
u=0,1,-1-i32,-1+i32
u=0,1,-1-i32,-1+i32
Substitute Q for u.
Q12=0,1,-1-i32,-1+i32
Solve for Q12=0 for Q.
Raise each side of the equation to the 2 power to eliminate the fractional exponent on the left side.
(Q12)2=(0)2
Raising 0 to any positive power yields 0.
Q=0
Q=0
Solve for Q12=1 for Q.
Raise each side of the equation to the 2 power to eliminate the fractional exponent on the left side.
(Q12)2=(1)2
One to any power is one.
Q=1
Q=1
Solve for Q12=-1-i32 for Q.
Raise each side of the equation to the 2 power to eliminate the fractional exponent on the left side.
(Q12)2=(-1-i32)2
Simplify (-1-i32)2.
Use the power rule (ab)n=anbn to distribute the exponent.
Apply the product rule to -1-i32.
Q=(-1)2(1-i32)2
Apply the product rule to 1-i32.
Q=(-1)2(1-i3)222
Q=(-1)2(1-i3)222
Simplify the expression.
Raise -1 to the power of 2.
Q=1(1-i3)222
Multiply (1-i3)222 by 1.
Q=(1-i3)222
Raise 2 to the power of 2.
Q=(1-i3)24
Rewrite (1-i3)2 as (1-i3)(1-i3).
Q=(1-i3)(1-i3)4
Q=(1-i3)(1-i3)4
Expand (1-i3)(1-i3) using the FOIL Method.
Apply the distributive property.
Q=1(1-i3)-i3(1-i3)4
Apply the distributive property.
Q=1⋅1+1(-i3)-i3(1-i3)4
Apply the distributive property.
Q=1⋅1+1(-i3)-i3⋅1-i3(-i3)4
Q=1⋅1+1(-i3)-i3⋅1-i3(-i3)4
Simplify and combine like terms.
Simplify each term.
Multiply 1 by 1.
Q=1+1(-i3)-i3⋅1-i3(-i3)4
Multiply -i3 by 1.
Q=1-i3-i3⋅1-i3(-i3)4
Multiply -1 by 1.
Q=1-i3-i3-i3(-i3)4
Multiply -i3(-i3).
Multiply -1 by -1.
Q=1-i3-i3+1i3(i3)4
Multiply 3 by 1.
Q=1-i3-i3+3i(i3)4
Raise 3 to the power of 1.
Q=1-i3-i3+313ii4
Raise 3 to the power of 1.
Q=1-i3-i3+3131ii4
Use the power rule aman=am+n to combine exponents.
Q=1-i3-i3+31+1ii4
Add 1 and 1.
Q=1-i3-i3+32ii4
Raise i to the power of 1.
Q=1-i3-i3+32(i1i)4
Raise i to the power of 1.
Q=1-i3-i3+32(i1i1)4
Use the power rule aman=am+n to combine exponents.
Q=1-i3-i3+32i1+14
Add 1 and 1.
Q=1-i3-i3+32i24
Q=1-i3-i3+32i24
Rewrite 32 as 3.
Use axn=axn to rewrite 3 as 312.
Q=1-i3-i3+(312)2i24
Apply the power rule and multiply exponents, (am)n=amn.
Q=1-i3-i3+312⋅2i24
Combine 12 and 2.
Q=1-i3-i3+322i24
Cancel the common factor of 2.
Cancel the common factor.
Q=1-i3-i3+322i24
Divide 1 by 1.
Q=1-i3-i3+31i24
Q=1-i3-i3+31i24
Evaluate the exponent.
Q=1-i3-i3+3i24
Q=1-i3-i3+3i24
Rewrite i2 as -1.
Q=1-i3-i3+3⋅-14
Multiply 3 by -1.
Q=1-i3-i3-34
Q=1-i3-i3-34
Subtract 3 from 1.
Q=-i3-i3-24
Subtract i3 from -i3.
Q=-2i3-24
Q=-2i3-24
Reorder -2i3 and -2.
Q=-2-2i34
Cancel the common factor of -2-2i3 and 4.
Factor 2 out of -2.
Q=2(-1)-2i34
Factor 2 out of -2i3.
Q=2(-1)+2(-i3)4
Factor 2 out of 2(-1)+2(-i3).
Q=2(-1-i3)4
Cancel the common factors.
Factor 2 out of 4.
Q=2(-1-i3)2⋅2
Cancel the common factor.
Q=2(-1-i3)2⋅2
Rewrite the expression.
Q=-1-i32
Q=-1-i32
Q=-1-i32
Rewrite -1 as -1(1).
Q=-1(1)-i32
Factor -1 out of -i3.
Q=-1(1)-(i3)2
Factor -1 out of -1(1)-(i3).
Q=-1(1+i3)2
Move the negative in front of the fraction.
Q=-1+i32
Q=-1+i32
Q=-1+i32
Solve for Q12=-1+i32 for Q.
Raise each side of the equation to the 2 power to eliminate the fractional exponent on the left side.
(Q12)2=(-1+i32)2
Simplify (-1+i32)2.
Use the power rule (ab)n=anbn to distribute the exponent.
Apply the product rule to -1+i32.
Q=(-1)2(1+i32)2
Apply the product rule to 1+i32.
Q=(-1)2(1+i3)222
Q=(-1)2(1+i3)222
Simplify the expression.
Raise -1 to the power of 2.
Q=1(1+i3)222
Multiply (1+i3)222 by 1.
Q=(1+i3)222
Raise 2 to the power of 2.
Q=(1+i3)24
Rewrite (1+i3)2 as (1+i3)(1+i3).
Q=(1+i3)(1+i3)4
Q=(1+i3)(1+i3)4
Expand (1+i3)(1+i3) using the FOIL Method.
Apply the distributive property.
Q=1(1+i3)+i3(1+i3)4
Apply the distributive property.
Q=1⋅1+1(i3)+i3(1+i3)4
Apply the distributive property.
Q=1⋅1+1(i3)+i3⋅1+i3(i3)4
Q=1⋅1+1(i3)+i3⋅1+i3(i3)4
Simplify and combine like terms.
Simplify each term.
Multiply 1 by 1.
Q=1+1(i3)+i3⋅1+i3(i3)4
Multiply i3 by 1.
Q=1+i3+i3⋅1+i3(i3)4
Multiply i by 1.
Q=1+i3+i3+i3(i3)4
Multiply i3(i3).
Raise i to the power of 1.
Q=1+i3+i3+i1i334
Raise i to the power of 1.
Q=1+i3+i3+i1i1334
Use the power rule aman=am+n to combine exponents.
Q=1+i3+i3+i1+1334
Add 1 and 1.
Q=1+i3+i3+i2334
Raise 3 to the power of 1.
Q=1+i3+i3+i2(313)4
Raise 3 to the power of 1.
Q=1+i3+i3+i2(3131)4
Use the power rule aman=am+n to combine exponents.
Q=1+i3+i3+i231+14
Add 1 and 1.
Q=1+i3+i3+i2324
Q=1+i3+i3+i2324
Rewrite i2 as -1.
Q=1+i3+i3-1324
Rewrite 32 as 3.
Use axn=axn to rewrite 3 as 312.
Q=1+i3+i3-1(312)24
Apply the power rule and multiply exponents, (am)n=amn.
Q=1+i3+i3-1⋅312⋅24
Combine 12 and 2.
Q=1+i3+i3-1⋅3224
Cancel the common factor of 2.
Cancel the common factor.
Q=1+i3+i3-1⋅3224
Divide 1 by 1.
Q=1+i3+i3-1⋅314
Q=1+i3+i3-1⋅314
Evaluate the exponent.
Q=1+i3+i3-1⋅34
Q=1+i3+i3-1⋅34
Multiply -1 by 3.
Q=1+i3+i3-34
Q=1+i3+i3-34
Subtract 3 from 1.
Q=i3+i3-24
Add i3 and i3.
Q=2i3-24
Q=2i3-24
Reorder 2i3 and -2.
Q=-2+2i34
Cancel the common factor of -2+2i3 and 4.
Factor 2 out of -2.
Q=2⋅-1+2i34
Factor 2 out of 2i3.
Q=2⋅-1+2(i3)4
Factor 2 out of 2⋅-1+2(i3).
Q=2⋅(-1+i3)4
Cancel the common factors.
Factor 2 out of 4.
Q=2⋅(-1+i3)2(2)
Cancel the common factor.
Q=2⋅(-1+i3)2⋅2
Rewrite the expression.
Q=-1+i32
Q=-1+i32
Q=-1+i32
Rewrite -1 as -1(1).
Q=-1(1)+i32
Factor -1 out of i3.
Q=-1(1)-(-i3)2
Factor -1 out of -1(1)-(-i3).
Q=-1(1-i3)2
Move the negative in front of the fraction.
Q=-1-i32
Q=-1-i32
Q=-1-i32
List all of the solutions.
Q=0,1,-1+i32,-1-i32
Solve for Q Q^(1/2)=Q^2

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