Algebra Examples

Solve by Factoring x^6-1=0
x6-1=0
Step 1
Rewrite x6 as (x2)3.
(x2)3-1=0
Step 2
Rewrite 1 as 13.
(x2)3-13=0
Step 3
Since both terms are perfect cubes, factor using the difference of cubes formula, a3-b3=(a-b)(a2+ab+b2) where a=x2 and b=1.
(x2-1)((x2)2+x21+12)=0
Step 4
Simplify.
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Step 4.1
Rewrite 1 as 12.
(x2-12)((x2)2+x21+12)=0
Step 4.2
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=x and b=1.
(x+1)(x-1)((x2)2+x21+12)=0
Step 4.3
Multiply x2 by 1.
(x+1)(x-1)((x2)2+x2+12)=0
(x+1)(x-1)((x2)2+x2+12)=0
Step 5
Simplify each term.
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Step 5.1
Multiply the exponents in (x2)2.
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Step 5.1.1
Apply the power rule and multiply exponents, (am)n=amn.
(x+1)(x-1)(x22+x2+12)=0
Step 5.1.2
Multiply 2 by 2.
(x+1)(x-1)(x4+x2+12)=0
(x+1)(x-1)(x4+x2+12)=0
Step 5.2
One to any power is one.
(x+1)(x-1)(x4+x2+1)=0
(x+1)(x-1)(x4+x2+1)=0
Step 6
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
x+1=0
x-1=0
x4+x2+1=0
Step 7
Set x+1 equal to 0 and solve for x.
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Step 7.1
Set x+1 equal to 0.
x+1=0
Step 7.2
Subtract 1 from both sides of the equation.
x=-1
x=-1
Step 8
Set x-1 equal to 0 and solve for x.
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Step 8.1
Set x-1 equal to 0.
x-1=0
Step 8.2
Add 1 to both sides of the equation.
x=1
x=1
Step 9
Set x4+x2+1 equal to 0 and solve for x.
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Step 9.1
Set x4+x2+1 equal to 0.
x4+x2+1=0
Step 9.2
Solve x4+x2+1=0 for x.
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Step 9.2.1
Substitute u=x2 into the equation. This will make the quadratic formula easy to use.
u2+u+1=0
u=x2
Step 9.2.2
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Step 9.2.3
Substitute the values a=1, b=1, and c=1 into the quadratic formula and solve for u.
-1±12-4(11)21
Step 9.2.4
Simplify.
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Step 9.2.4.1
Simplify the numerator.
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Step 9.2.4.1.1
One to any power is one.
u=-1±1-41121
Step 9.2.4.1.2
Multiply -411.
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Step 9.2.4.1.2.1
Multiply -4 by 1.
u=-1±1-4121
Step 9.2.4.1.2.2
Multiply -4 by 1.
u=-1±1-421
u=-1±1-421
Step 9.2.4.1.3
Subtract 4 from 1.
u=-1±-321
Step 9.2.4.1.4
Rewrite -3 as -1(3).
u=-1±-1321
Step 9.2.4.1.5
Rewrite -1(3) as -13.
u=-1±-1321
Step 9.2.4.1.6
Rewrite -1 as i.
u=-1±i321
u=-1±i321
Step 9.2.4.2
Multiply 2 by 1.
u=-1±i32
u=-1±i32
Step 9.2.5
Simplify the expression to solve for the + portion of the ±.
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Step 9.2.5.1
Simplify the numerator.
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Step 9.2.5.1.1
One to any power is one.
u=-1±1-41121
Step 9.2.5.1.2
Multiply -411.
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Step 9.2.5.1.2.1
Multiply -4 by 1.
u=-1±1-4121
Step 9.2.5.1.2.2
Multiply -4 by 1.
u=-1±1-421
u=-1±1-421
Step 9.2.5.1.3
Subtract 4 from 1.
u=-1±-321
Step 9.2.5.1.4
Rewrite -3 as -1(3).
u=-1±-1321
Step 9.2.5.1.5
Rewrite -1(3) as -13.
u=-1±-1321
Step 9.2.5.1.6
Rewrite -1 as i.
u=-1±i321
u=-1±i321
Step 9.2.5.2
Multiply 2 by 1.
u=-1±i32
Step 9.2.5.3
Change the ± to +.
u=-1+i32
Step 9.2.5.4
Rewrite -1 as -1(1).
u=-11+i32
Step 9.2.5.5
Factor -1 out of i3.
u=-11-(-i3)2
Step 9.2.5.6
Factor -1 out of -1(1)-(-i3).
u=-1(1-i3)2
Step 9.2.5.7
Move the negative in front of the fraction.
u=-1-i32
u=-1-i32
Step 9.2.6
Simplify the expression to solve for the - portion of the ±.
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Step 9.2.6.1
Simplify the numerator.
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Step 9.2.6.1.1
One to any power is one.
u=-1±1-41121
Step 9.2.6.1.2
Multiply -411.
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Step 9.2.6.1.2.1
Multiply -4 by 1.
u=-1±1-4121
Step 9.2.6.1.2.2
Multiply -4 by 1.
u=-1±1-421
u=-1±1-421
Step 9.2.6.1.3
Subtract 4 from 1.
u=-1±-321
Step 9.2.6.1.4
Rewrite -3 as -1(3).
u=-1±-1321
Step 9.2.6.1.5
Rewrite -1(3) as -13.
u=-1±-1321
Step 9.2.6.1.6
Rewrite -1 as i.
u=-1±i321
u=-1±i321
Step 9.2.6.2
Multiply 2 by 1.
u=-1±i32
Step 9.2.6.3
Change the ± to -.
u=-1-i32
Step 9.2.6.4
Rewrite -1 as -1(1).
u=-11-i32
Step 9.2.6.5
Factor -1 out of -i3.
u=-11-(i3)2
Step 9.2.6.6
Factor -1 out of -1(1)-(i3).
u=-1(1+i3)2
Step 9.2.6.7
Move the negative in front of the fraction.
u=-1+i32
u=-1+i32
Step 9.2.7
The final answer is the combination of both solutions.
u=-1-i32,-1+i32
Step 9.2.8
Substitute the real value of u=x2 back into the solved equation.
x2=-1-i32
(x2)1=-1+i32
Step 9.2.9
Solve the first equation for x.
x2=-1-i32
Step 9.2.10
Solve the equation for x.
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Step 9.2.10.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
x=±-1-i32
Step 9.2.10.2
Simplify ±-1-i32.
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Step 9.2.10.2.1
Rewrite -1-i32 as i212-i32.
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Step 9.2.10.2.1.1
Rewrite -1 as i2.
x=±i21-i32
Step 9.2.10.2.1.2
Rewrite 1 as 12.
x=±i212-i32
x=±i212-i32
Step 9.2.10.2.2
Pull terms out from under the radical.
x=±i12-i32
Step 9.2.10.2.3
One to any power is one.
x=±i1-i32
Step 9.2.10.2.4
Rewrite 1-i32 as 1-i32.
x=±i1-i32
Step 9.2.10.2.5
Multiply 1-i32 by 22.
x=±i(1-i3222)
Step 9.2.10.2.6
Combine and simplify the denominator.
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Step 9.2.10.2.6.1
Multiply 1-i32 by 22.
x=±i1-i3222
Step 9.2.10.2.6.2
Raise 2 to the power of 1.
x=±i1-i32212
Step 9.2.10.2.6.3
Raise 2 to the power of 1.
x=±i1-i322121
Step 9.2.10.2.6.4
Use the power rule aman=am+n to combine exponents.
x=±i1-i3221+1
Step 9.2.10.2.6.5
Add 1 and 1.
x=±i1-i3222
Step 9.2.10.2.6.6
Rewrite 22 as 2.
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Step 9.2.10.2.6.6.1
Use axn=axn to rewrite 2 as 212.
x=±i1-i32(212)2
Step 9.2.10.2.6.6.2
Apply the power rule and multiply exponents, (am)n=amn.
x=±i1-i322122
Step 9.2.10.2.6.6.3
Combine 12 and 2.
x=±i1-i32222
Step 9.2.10.2.6.6.4
Cancel the common factor of 2.
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Step 9.2.10.2.6.6.4.1
Cancel the common factor.
x=±i1-i32222
Step 9.2.10.2.6.6.4.2
Rewrite the expression.
x=±i1-i3221
x=±i1-i3221
Step 9.2.10.2.6.6.5
Evaluate the exponent.
x=±i1-i322
x=±i1-i322
x=±i1-i322
Step 9.2.10.2.7
Combine using the product rule for radicals.
x=±i(1-i3)22
Step 9.2.10.2.8
Combine i and (1-i3)22.
x=±i(1-i3)22
Step 9.2.10.2.9
Move 2 to the left of 1-i3.
x=±i2(1-i3)2
x=±i2(1-i3)2
Step 9.2.10.3
The complete solution is the result of both the positive and negative portions of the solution.
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Step 9.2.10.3.1
First, use the positive value of the ± to find the first solution.
x=i2(1-i3)2
Step 9.2.10.3.2
Next, use the negative value of the ± to find the second solution.
x=-i2(1-i3)2
Step 9.2.10.3.3
The complete solution is the result of both the positive and negative portions of the solution.
x=i2(1-i3)2,-i2(1-i3)2
x=i2(1-i3)2,-i2(1-i3)2
x=i2(1-i3)2,-i2(1-i3)2
Step 9.2.11
Solve the second equation for x.
(x2)1=-1+i32
Step 9.2.12
Solve the equation for x.
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Step 9.2.12.1
Remove parentheses.
x2=-1+i32
Step 9.2.12.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
x=±-1+i32
Step 9.2.12.3
Simplify ±-1+i32.
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Step 9.2.12.3.1
Rewrite -1+i32 as i212+i32.
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Step 9.2.12.3.1.1
Rewrite -1 as i2.
x=±i21+i32
Step 9.2.12.3.1.2
Rewrite 1 as 12.
x=±i212+i32
x=±i212+i32
Step 9.2.12.3.2
Pull terms out from under the radical.
x=±i12+i32
Step 9.2.12.3.3
One to any power is one.
x=±i1+i32
Step 9.2.12.3.4
Rewrite 1+i32 as 1+i32.
x=±i1+i32
Step 9.2.12.3.5
Multiply 1+i32 by 22.
x=±i(1+i3222)
Step 9.2.12.3.6
Combine and simplify the denominator.
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Step 9.2.12.3.6.1
Multiply 1+i32 by 22.
x=±i1+i3222
Step 9.2.12.3.6.2
Raise 2 to the power of 1.
x=±i1+i32212
Step 9.2.12.3.6.3
Raise 2 to the power of 1.
x=±i1+i322121
Step 9.2.12.3.6.4
Use the power rule aman=am+n to combine exponents.
x=±i1+i3221+1
Step 9.2.12.3.6.5
Add 1 and 1.
x=±i1+i3222
Step 9.2.12.3.6.6
Rewrite 22 as 2.
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Step 9.2.12.3.6.6.1
Use axn=axn to rewrite 2 as 212.
x=±i1+i32(212)2
Step 9.2.12.3.6.6.2
Apply the power rule and multiply exponents, (am)n=amn.
x=±i1+i322122
Step 9.2.12.3.6.6.3
Combine 12 and 2.
x=±i1+i32222
Step 9.2.12.3.6.6.4
Cancel the common factor of 2.
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Step 9.2.12.3.6.6.4.1
Cancel the common factor.
x=±i1+i32222
Step 9.2.12.3.6.6.4.2
Rewrite the expression.
x=±i1+i3221
x=±i1+i3221
Step 9.2.12.3.6.6.5
Evaluate the exponent.
x=±i1+i322
x=±i1+i322
x=±i1+i322
Step 9.2.12.3.7
Combine using the product rule for radicals.
x=±i(1+i3)22
Step 9.2.12.3.8
Combine i and (1+i3)22.
x=±i(1+i3)22
Step 9.2.12.3.9
Move 2 to the left of 1+i3.
x=±i2(1+i3)2
x=±i2(1+i3)2
Step 9.2.12.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 9.2.12.4.1
First, use the positive value of the ± to find the first solution.
x=i2(1+i3)2
Step 9.2.12.4.2
Next, use the negative value of the ± to find the second solution.
x=-i2(1+i3)2
Step 9.2.12.4.3
The complete solution is the result of both the positive and negative portions of the solution.
x=i2(1+i3)2,-i2(1+i3)2
x=i2(1+i3)2,-i2(1+i3)2
x=i2(1+i3)2,-i2(1+i3)2
Step 9.2.13
The solution to x4+x2+1=0 is x=i2(1-i3)2,-i2(1-i3)2,i2(1+i3)2,-i2(1+i3)2.
x=i2(1-i3)2,-i2(1-i3)2,i2(1+i3)2,-i2(1+i3)2
x=i2(1-i3)2,-i2(1-i3)2,i2(1+i3)2,-i2(1+i3)2
x=i2(1-i3)2,-i2(1-i3)2,i2(1+i3)2,-i2(1+i3)2
Step 10
The final solution is all the values that make (x+1)(x-1)(x4+x2+1)=0 true.
x=-1,1,i2(1-i3)2,-i2(1-i3)2,i2(1+i3)2,-i2(1+i3)2
x6-1=0
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