Question

Without solving the equation 3x^2−12x+7=0 find the sum of the squares of its roots.

Answer 1

`(34)/(3)`

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 +
`(bx)/(a)` +
`(c)/(a)` = 0 ) then

α + β = -
`(b)/(a)` , αβ =
`(c)/(a)`

comparing 3x² - 12x + 7 = 0 with ax² + bx + c = 0, gives

a = 3, b = - 12, c = 7, hence

α + β = -
`(-12)/(3)` = 4 and αβ =
`(7)/(3)`

(α + β)² = α² + β² + 2αβ

α² + β² = (α + β)² - 2αβ = 4² -
`(14)/(3)` =
`(34)/(3)`