Question

if the difference of the roots of quadratic equation ax^2+2hx+b=0 is 1,then prove that : 4(h^2-ab) = a^2​

Answer 1

Answer:

The given quadratic equation is;

ax^2 + 2hx + b = 0.

Let A and B are the roots of given polynomial.

Also, it is given that;

The difference of the roots of the given equation is 1.

Thus;

A - B = 1

Also,

We know that,

The sum of both the roots of a quadratic equation in variable x is given by;

-(coefficient of x)/(coefficient of x^2)

Thus;

A + B = -2h/a

Also,

The product of both the roots of a quadratic equation in variable x is given by;

(constant term)/(coefficient of x^2)

Thus;

A•B = b/a

Also,

We know that;

(A+B)^2 = (A-B)^2 + 4•A•B

Now,

Putting the appropriate values of (A+B),(A-B) and A•B in above formula,

We get;

=> (A+B)^2 = (A-B)^2 + 4•A•B

=> (-2h/a)^2 = (1)^2 + 4•(b/a)

=> 4h^2/a^2 = 1 + 4b/a

=> 4h^2 = a^2(1 + 4b/a)

=> 4h^2 = a^2 + 4ab

=> 4h^2 - 4ab = a^2

=> 4(h^2 - ab) = a^2

Hence proved.