If the roots of \[x^2 - bx + c = 0\] are two consecutive integers, then b2 − 4 c is
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Solution
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Given equation:
\[x^2 - bx + c = 0\]
Let \[\alpha \text { and } \alpha + 1\] be the two consecutive roots of the equation.
Sum of the roots = \[\alpha + \alpha + 1 = 2\alpha + 1\]
Product of the roots = \[\alpha (\alpha + 1) = \alpha^2 + \alpha\]
\[\text { So, sum of the roots }= 2\alpha + 1 = \frac{- \text { Coeffecient of } x}{\text { Coeffecient of } x^2} = \frac{b}{1} = b\]
\[\text { Product of the roots } = \alpha^2 + \alpha = \frac{\text { Constant term }}{\text { Coeffecient of }x^2} = \frac{c}{1} = c\]
\[\text { Now, } b^2 - 4c = \left( 2\alpha + 1 \right)^2 - 4\left( \alpha^2 + \alpha \right) = 4 \alpha^2 + 4\alpha + 1 - 4 \alpha^2 - 4\alpha = 1\]
