suppose α and β are the roots of a quadratic equation. It is known that α² + β² =9 and (α+3β)(3α+β) =-13.?

Suppose α and β are the roots of a quadratic equation. It is known that α^2 + β^2=9 and (α+3β)(3α+β)=－13
a) Find the quadratic equation
Sol
(α+3β)(3α+β)=－13
3α^2+10αβ+3β^2=－13
10αβ=－13－27=－40
αβ=－4
(α+β)^2－2αβ=9
(α+β)^2=1
α+β=1 or α+β=－1
(1) α+β=1
x^2-(α+β)x+αβ=0
x^2－x－4=0………….
(2) α+β=－1
x^2－(α+β)x+αβ=0
x^2+x+4=0……………
b) Form a quadratic equation with roots α^2+1 and β^2+1
Sol
(α^2+1)+( β^2+1)=11
(α^2+1)( β^2+1)= α^2β^2+(α^2+β^2)+1=16+9+1=26
x^2－11x+26=0