Question

if the roots of the equation x^2 +ax-b=0 differ by unity.prove a^2 +4b-1=0

Answer 1

The proof has been shown in the explanation part.

Step-by-step explanation:

The given quadratic equation is $x^2+ax-b$

Let the one root of this equation is y then other root must be y +1

Now, sum of roots is given by $-\frac{b}{a}$

$y+y+1=-a\\\\2y=-a-1\\\\y=\frac{1}{2}(-a-1)$

Now, product of roots is given by $\frac{c}{a}$

$y(y+1)=-b\\\\y^2+y=-b$

Substituting the value of y

$(\frac{1}{2}(-a-1))^2+\frac{1}{2}(-a-1)=-b\\\\==\frac{\left(-a-1\right)^2+\left(-a-1\right)\cdot \:2}{4}=-b\\\\a^2+2a+1-2a-2=-4b\\\\a^2+4b-1=0$

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