quadratic Equations

Question

Without solving the equation 5x2+13x−6=0 find the sum of the squares of its roots.

Without solving the equation 5x²+13x−6=0 find the sum of the squares of its roots.

Answer 1

Take the positive case of the quadratic formula and square it.

 

  [(-B + square-root(B^2-4AC)/(2*A)]^2 =

 

   [b^2 + 2B*square-root(B^2-4AC) + B^2-4AC ] /4A^2

 

  = [2*B^2 + 2B*square-root(B^2-4AC)  - 4Ac]/4A^2

 

  

Then take the negative case of the quadratic formula and square it.

 

That part is [2*B^2 - 2B*square-root(B^2-4AC) - 4Ac]/4A^2 

 

Then adding those piecces together and doing the algebra, the sum of the squares of the two

parts of the quadratic formula:

 

  (B^2 - 2AC)/A^2

 

  In this case, A=5, B=13, C = -6

 

   so the sum of the squares of the roots per the derived formula is  (13^2 - 2*5*(-6))/5^2

 

                                                                                             = (169 + 60)/25 = 229/25

 

 Now I am going to do what you told me not to do just to check it.

 Solving by factoring ( 5x   -    2)( x     +     3  )

  so the roots are 2/5 and -3.

 their squares are 4/25 and 9

 the total is 9 and 4/25 = 229/25

 which verifies the answer.