SOLUTION: if a^2 + 1 = a find the value of a^3.

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Question 1087364: if a^2 + 1 = a find the value of a^3.
Found 4 solutions by jim_thompson5910, Edwin McCravy, MathTherapy, ikleyn:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Start with the given equation

Subtract 1 from both sides

Simplify. Call this equation (1)

Go back to the original equation

Multiply both sides by 'a'

Distribute

Subtract 'a' from both sides to isolate the a^3 term

Simplfy

Replace the with . See equation (1) above.

Simplify. Note how the 'a' terms on the right side cancel out

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An alternative is to solve for 'a' using the quadratic formula (after you get everything to one side and have ). That leads to two complex solutions in the form . After cubing both sides of the two equations you'll lead to . This method is a lot longer than the first method so I find it ideal to follow the first method.

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No matter which method you follow, the final answer is -1
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
 

if a^2 + 1 = a find the value of a^3.

    a² + 1 = a

a² - a + 1 = 0

We recognize the left side as one of the factors in
the factorization of the sum of two cubes 

a³ + 1 = (a + 1)(a² - a + 1) 

So we multiply both sides by (a + 1)

(a + 1)(a² - a + 1) = 0(a + 1)

             a³ + 1 = 0
                 
                 a³ = -1

                

Edwin
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

if a^2 + 1 = a find the value of a^3.




 ------- Multiplying each side by a

  

 

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
 .


What is interesting in this problem (and why I am writing these lines) 

is that  = -1 ,  BUT  a =/= -1  (!!).



The value of  "a"  in this problem is one of the two complex cubic roots of -1.





 

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