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Let a and b be the roots of the quadratic x^2 - 5x + 3 = 2x^2 + 15x - 10. Find the quadratic whose roots are a^2 + a + 1 and b^2 + b + 1.

 Feb 15, 2024
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Simplify as

x^2 + 20x - 13 =  0

 

Product of the  roots = ab = -13      (ab)^2  = 169

2ab = -26

 

Sum of the  roots  = a + b =  -20

(a + b)^2  = (-20)^2

a^2 + 2ab + b^2  = 400

a^2 + b^2  + 2ab = 400

a^2 + b^2  - 26  = 400

a^2 + b^2 =  426

 

For  the new quadratic,  the  sum of the roots =  (a^2 + b^2) + (a + b) + 2 =  426 - 20 + 2  =  408

 

The product of the roots = (a^2 + a + 1) (b^2 + b + 1)   =

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

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

426 + 169 + (-13)(-20) -13 - 20 + 1  = 823

 

The   quadratic is  Ax^2 + Bx + C  and we  can let A = 1

 

x^2 - 408x + 823

 

cool cool cool

 Feb 16, 2024

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