First slide

Theory of expressions

Question

If the roots of the equation $x^{2} - p x + q = 0$ differ by unity, then

Moderate

A

$p^{2} = 4 q$
B

$p^{2} = 4 q + 1$
C

$p^{2} = 4 q - 1$
D

none of these

Solution

Let the roots be $a$ and $a + 1$ . Then,
$α + α + 1 = p \Rightarrow α = \frac{p - 1}{2}$ (i..)
and , $α (α + 1) = q \Rightarrow α^{2} + α = q$ (ii…)
From (i) and (ii), we get
${(\frac{p - 1}{2})}^{2} + (\frac{p - 1}{2}) = q$ [On eliminating $a$ ]
$\Rightarrow p^{2} - 2 p + 1 + 2 p - 2 = 4 q \Rightarrow p^{2} = 4 q + 1$ .
ALTER Let the roots be $α$ and $β$ . Then,
$| α - β | = 1 \Rightarrow (α - β)^{2} = 1 \Rightarrow (α + β)^{2} - 4 α β = 1 \Rightarrow p^{2} - 4 q = 1$

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