Answer:
[tex]\frac{34}{3}[/tex]
Step-by-step explanation:
If α + β are the roots of the equation ax² + bx + c = 0 then
(x - α)(x - β) = 0, that is
x² - x(α + β) + αβ = 0
comparing the equation with ax² + bx + c = 0 ( ie. x² + [tex]\frac{bx}{a}[/tex] + [tex]\frac{c}{a}[/tex] = 0 ) then
α + β = - [tex]\frac{b}{a}[/tex] , αβ = [tex]\frac{c}{a}[/tex]
comparing 3x² - 12x + 7 = 0 with ax² + bx + c = 0, gives
a = 3, b = - 12, c = 7, hence
α + β = - [tex]\frac{-12}{3}[/tex] = 4 and αβ = [tex]\frac{7}{3}[/tex]
(α + β)² = α² + β² + 2αβ
α² + β² = (α + β)² - 2αβ = 4² - [tex]\frac{14}{3}[/tex] = [tex]\frac{34}{3}[/tex]
