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Question

If α and β are the roots of the equation x2 – 5x + 6 = 0, find

α2 + β2

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Solution

Given: x2 – 5x + 6 = 0
On comparing this equation with ax2 + bx + c = 0, we get:
a = 1, b = –5 and c = 6
α and β are the roots of the equation.
To find the value of α2 + β2, we need to express it in terms of (α + β) and αβ because we know that α + β = –ba and αβ = ca.
Thus,
α2 + β2 = α2 + β2 + 2αβ – 2αβ
α2 + β2 = (α + β)2 – 2αβ
On substituting α + β = –ba and αβ = ca, we get:
α2 + β2 = -ba2 -2 ×ca = b2 -2caa2
On substituting a = 1, b = –5 and c = 6, we get:

α2 + β2 = -52-2×6×112 =25-121 =13

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