# If alpha and beta are the zeros of x^2-2x+3, find a quadratic polynomial whose roots are alpha+beta , beta+2

2
by Sivesh

2015-04-12T20:16:22+05:30
Given that alpha & beta are the roots of the quadratic equation x^2-2x+3
Comparing it with ax^2+ bx+c, we have a=1, b= -2 & c=3
Therefore, alpha+beta = -(b)/a = -(-2)/1 = 2
& Alpha*Beta = c/a = 3/1 = 3
Now, (alpha+2)+(beta+2) = (alpha+beta)+4 = (2)+4 = 6
& (alpha+2)*(beta+2) = alpha*beta +2(alpha+beta)+4 = 3+2(2)+4 = 3+4+4=11
Hence, the required polynomial is x^2-6x+11
Then if the roots are alpha+beta and beta+2
2015-04-12T22:40:29+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
f(x) = x² - p x + q
α and β are the roots of the above equation.
to find   α² / β² +  β² / α² = ?

α = [ p + √(p² - 4q)  ]  / 2        and    β = [ p - √(p² - 4 q) ] / 2

so,      α + β = p      and    α β  =  q          and     α² β²  = q²

=>    α² + β²   =  (α+β)² - 2 αβ  =  p² - 2 q

=>    α⁴ + β⁴  =  (α² + β²)² - 2 α²β²  =  (p² - 2q)² - 2 q²
=  p⁴ - 4 p² q + 4 q² - 2 q²
=  p⁴ - 4 p² q + 2 q²

NOW ,  α² / β² + β² / α² =  [ α⁴  + β⁴ ] / α² β² =
= [ p⁴ - 4 p² q + 2 q² ] /  q²