Question

If α,β are the roots of 4x²+7x+2=0 then the equation whose roots are α²,β² is ??????????

Answer 1

so we know that for quadratic equation ax² +bx +c = 0
whose roots are m,n we know that x² - ( m + n)x +mn = 0 the previous quadratic equation
we know sum of roots = -b/a
product of roots = c/a
so in this sum
roots of the equation are 4x² +7x + 2 = 0
                  α +β  = -7/4
                    αβ = 2/4 = 1/2
 so to form the quadratic equation as given above using x² - ( m + n)x +mn = 0
so x² - ( α² + β² )x + (αβ)² = 0
so using formula a² +b² = (a +b)² - 2ab

so x² - ( α² + β² )x + (αβ)² = 0

⇒x² - {(α+ β)² - 2αβ }x + (αβ)² = 0
⇒x² - {(-7/2)² - 2 ×1/2}x + (1/2)² = 0
⇒4x²  - 49x + 1 = 0
so the required quadratic equation is 

4x²  - 49x + 1 = 0