Question

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

α² + β²

Answer 1

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 = –$\frac{b}{a}$ and αβ = $\frac{c}{a}$.
Thus,
α² + β² = α² + β² + 2αβ – 2αβ
$\Rightarrow$ α² + β² = (α + β)² – 2αβ
On substituting α + β = –$\frac{b}{a}$ and αβ = $\frac{c}{a}$, we get:
$\Rightarrow$ α² + β² = ${\left(\frac{-b}{a}\right)}^2-2\times \frac{c}{a}=\frac{b^2-2ca}{a^2}$
On substituting a = 1, b = –5 and c = 6, we get:

$$\begin{gathered} {\alpha }^2+{\beta }^2=\frac{{\left(-5\right)}^2-2\times 6\times 1}{1^2} \\ =\frac{25-12}{1} \\ =13 \end{gathered}$$