Question

if the roots of a^2 x^2 - ab x + c = 0 are consecutive integers then b^2 - a^2​

Answer 1

From the given picture,

Difference between the roots of a quadratic equation : $\dfrac{\sqrt{b^2 - 4ac}}{a}$

Let the required consecutive integers are k and k + 1.

Thus, difference between the roots :

→ [ √( b^2 - 4ac ) / a ] = difference between roots. [ if equation is ax^2 + bx + c = 0 ]

On comparing the given equation with ax^2 + bx + c = 0, we get a = a^2, b = ab, c = c. [ variables of the quadratic ax^2 + bx + c = 0 are different from the variables of given equation. ]

= > [ √{ ( - ab )^2 - 4( a^2 c ) } / a^2 ] = k + 1 - k

= > [ √{ ( - ab )^2 - 4( a^2 c ) } / a^2 ] = 1

= > √( a^2 b^2 - 4a^2 c ) = a^2

= > a^2 b^2 - 4 a^2 c = a^4

= > a^2 ( b^2 - 4 c ) = a^4

= > b^2 - 4c = a^2

= > b^2 - a^2 = 4c

Hence the required value of b^2 - a^2 is 4c.