If the difference of the roots of the quadratic equation [tex]x^2 - px + q = 0[/tex] is unity, then which of the following is true?
A. [tex]p^2 - 4q = 1[/tex]
B. [tex]4p^2 + q^2 = (1 + 2p)^2[/tex]
C. [tex]p^2 + 4q^2 = (1 + 2q)^2[/tex]
D. Both (A) and (C)
Answer (2)
The difference of the roots of the quadratic equation leads to the conclusion that p 2 − 4 q = 1 , which corresponds to option A. Other options either do not hold or cannot be derived from the given information. Thus, the correct choice is A.
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To solve the problem, we start by considering the properties of the roots of the quadratic equation.
For the quadratic equation x 2 − p x + q = 0 , let the roots be α and β .
According to Vieta's formulas, the sum of the roots α + β is equal to p and the product of the roots α β is equal to q .
It is given that the difference of the roots is unity, i.e., ∣ α − β ∣ = 1 .
To address this condition, we use the formula for the difference of roots of a quadratic equation:
∣ α − β ∣ = ( α + β ) 2 − 4 α β = p 2 − 4 q
Given ∣ α − β ∣ = 1 , we equate:
p 2 − 4 q = 1
Squaring both sides, we obtain:
p 2 − 4 q = 1
This is option (A).
Let's verify by evaluating our options:
(A) p 2 − 4 q = 1 : This is the result we derived, so it's true.
(B) 4 p 2 + q 2 = ( 1 + 2 p ) 2 : We did not derive this from our calculations, so it isn't confirmed by our logic here.
(C) p 2 + 4 q 2 = ( 1 + 2 q ) 2 : Similarly, this does not align with our derived condition p 2 − 4 q = 1 .
From the above, the answer is clearly:
Option A: p 2 − 4 q = 1 .
