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Question
If the difference between the roots of the equation \[x^2 + ax + 8 = 0\] is 2, write the values of a.
Solution
Given:
\[x^2 + ax + 8 = 0 .\]
Let \[\alpha \text { and } \beta\] are the roots of the equation.
Sum of the roots = \[\alpha + \beta = \frac{- a}{1} = - a\].
Product of the roots = \[\alpha\beta = \frac{8}{1} = 8\]
Given:
\[\alpha - \beta = 2\]
\[\text { Then }, \left( \alpha + \beta \right)^2 - \left( \alpha - \beta \right)^2 = 4\alpha\beta\]
\[ \Rightarrow \left( \alpha + \beta \right)^2 - 2^2 = 4 \times 8\]
\[ \Rightarrow \left( \alpha + \beta \right)^2 - 4 = 32\]
\[ \Rightarrow \left( \alpha + \beta \right)^2 = 32 + 4 = 36\]
\[ \Rightarrow \left( \alpha + \beta \right) = \pm 6\]
\[\alpha + \beta = - a = \pm 6\]
\[a = \pm 6\]
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