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avatar+1593 

Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = x^2 + 15x + 23.  Compute a^4 + b^4.

 Feb 15, 2024
 #1
avatar+157 
+2

Let a and b be the roots of the quadratic 2x28x+7=x2+15x+23.  Compute a4+b4.

 

We can first calculate the roots of the quadratic.


They are: 

 

23+5932 and 235932.

 

Now we can calculate the a4+b4 part.

 

(23+5932)4+(235932)4

=314209+129035932+314209129035932

=314209

 

Our final answer is 314209.

 

*There is probably a much faster and more efficient way to solve this that someone more educated than I am will be able to teach you clearly blush

 Feb 16, 2024
edited by BlackjackEd  Feb 16, 2024
 #2
avatar+130194 
+1

Thx, BlackjackEd....here's another way without having to actually calculate the roots

 

Simplify as

x^2 - 23x - 16  =  0

 

Product of the roots =  ab  = -16

2ab = -32

2(a^2b^2)  =  2(ab)^2  = 2(-16)^2  = 512

 

Sum of the  roots =  a + b = 23

(a + b)^2  = 529

(a^2 + 2ab + b^2) = 529

(a^2 + b^2)  + 2ab = 529

(a^2 + b^2) + (-32) = 529

(a^2 + b^2) -  32 = 529

(a^2 + b^2)  = 529  + 32   =  561

 

(a^2 + b^2)^2  =  a^4 + 2a^2b^2  + b^4

(a^2 + b^2)^2 - 2a^2b^2  = a^4 + b^4

(561)^2  - 512  =  a^4 + b^4

314209  = a^4 + b^4

 

cool cool cool

 Feb 16, 2024

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