If `m(ax^2+2bx+c)+px^2+2qx+r` can be expressed in the form `n(x+k)^2` then show that `(ak-b)(qk-r) =(pk-q)(bk-c).`

If p, q, r are rational then show that x^(2)-2px+p^(2)-q^(2)+2qr-r^(2)=0

If the equations ax^2 + bx + c = 0 and x^3 + x - 2 = 0 have two common roots then show that 2a = 2b = c .

If the roots of ax^2+2bx+c=0 be possible and different then show that the roots of (a+c)(ax^2+2bx+2c)=2(ac-b^2)(x^2+1) will be impossible and vice versa

The sum of −2x^2 + x + 31 and 3x^2 + 7x − 8 can be written in the form ax^2 + bx + c , where a, b, and c are constants.What is the value of a + b + c ?

If r is the ratio of the roots of the equation ax^2 + bx + c = 0 , show that (r+1)^2/r =(b^2)/(ac)

If a,b,c and d are in G.P , then show that ax^(2)+c divides ax^(3)+bx^(2)+cx+d

If a,b,c are the roots of the equation x^3+px^2+qx+r=0 such that c^2=-ab show that (2q-p^2)^3. r=(pq-4r)^3 .

(-3x^(2) + 5x -2) -2(x^(2)-2x-1) If the expression above is rewritten in the form ax^(2) + bx +c , where a, b, and c are constants, what is the value of b ?

If the sum of the rotsof the equation px^2+qx+r=0 be equal to the sum of their squares, show that 2pr=pq+q^2

If a, b, c ∈ R, a ≠ 0 and the quadratic equation ax^2 + bx + c = 0 has no real root, then show that (a + b + c) c > 0