Multiple choice

State the following statement is True or False If the roots of the equation $x^2\,+\,px\,+\,q\,=\,0$ differ by $1$, then $p^2\,=\,1\,+\,4q$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If roots are r and r+1, then sum = 2r+1 = -p and product = r(r+1) = q. Then p^2 = (2r+1)^2 = 4r^2 + 4r + 1. Since 4q = 4r^2 + 4r, p^2 = 4q + 1.

AI explanation

Let the roots of the quadratic equation x^2 + px + q = 0 be alpha and beta, where their difference is 1. We can write this relationship as (alpha - beta)^2 = 1^2 = 1. Using the standard identity relating the roots and coefficients, we know (alpha - beta)^2 = (alpha + beta)^2 - 4*alpha*beta. Since alpha + beta = -p and alpha*beta = q, substituting these values gives 1 = p^2 - 4q. Rearranging this equation results in p^2 = 1 + 4q, making the statement true.

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