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Let a and b be the roots of the quadratic $x^2 - 5x + 3 = 4x^2 - 8x + 23$ Find the quadratic whose roots are a^4 and b^4.

 Mar 16, 2024
 #1
avatar+128406 
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Simplify as

 

3x^2 -3x + 20   =    0 

 

Product of roots =  ab  = 20/3

2ab = 40/3

 

Sum of  roots =   a + b  =   - (-3) / 3  = 1

Square both sides

a^2 + 2ab + b^2 = 1

a^2 + b^2  + 40/3 = 1

a^2 + b^2  = -37/3               square both sides

 

a^4 + 2(ab)^2  + b^4  =  1369 /  9

a^4 + b^4  + 2(20/3)^2  = 1369 /  9

a^4 + b^4 + 800/9  = 1369/9

a^4 + b^4  = 569 / 9

 

 

cool cool cool

 Mar 16, 2024

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