First slide

Theory of equations

Question

If the roots of $x^{2} - bx + c = 0$ are two consecutive integers, then $b^{2} - 4 c$ is

Moderate

A

1
B

2
C

3
D

4

Solution

Let the roots are $α$ and $α + 1,$
then, sum of roots $= 2 α + 1 = b$
product of roots $= α (α + 1) = c$
Now, $b^{2} - 4 c = {(2 α + 1)}^{2} - 4 α (α + 1)$
$= 4 α^{2} + 1 + 4 α - 4 α^{2} - 4 α$
$∴ b^{2} - 4 c = 1$

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