`x_1` and `x_2` are the roots of `a x^2+b x+c=0` and `x_1x_2<0.` Roots of `x_1(x-x_2)^2+x_2(x-x_1)^2=0` are: (a) real and of opposite sign b. negative c. positive d. non real

Find the condition if the roots of a x^2+2b x+c=0 and b x^2-2sqrt(a c )x+b=0 are simultaneously real.

If x_1, a n dx_2 are the roots of x^2+(sintheta-1)x-1/2(cos^2theta)=0, then find the maximum value of x_1^2+x_2^2

If x=1 and x=2 are solutions of equations x^3+a x^2+b x+c=0 and a+b=1, then find the value of bdot

If x_(1) and x_(2) are the real and distinct roots of ax^(2)+bx+c=0 then prove that lim_(xtox1) (1+sin(ax^(2)+bx+c))^(1/(x-x_(1)). equals to

If alpha,beta are the roots of the equation a x^2+b x+c=0, then find the roots of the equation a x^2-b x(x-1)+c(x-1)^2=0 in term of alpha and betadot

Find the roots of the equation 2x^2 - x + 1/8 = 0

Find the condition so that the two equations a_1 x^2+b_1 x+c_1=0 and a_2 x^2+b_2 x+c_2=0 will have a common root.

If x_1 and x_2 are two distinct roots of the equation acosx+bsinx=c , then tan((x_1+x_2)/2) is equal to (a) a/b (b) b/a (c) c/a (d) a/c

If alpha and beta are roots of the equation a x^2+b x+c=0, then the roots of the equation a(2x+1)^2-b(2x+1)(x-3)+c(x-3)^2=0 are a. (2alpha+1)/(alpha-3),(2beta+1)/(beta-3) b. (3alpha+1)/(alpha-2),(3beta+1)/(beta-2) c. (2alpha-1)/(alpha-2),(2beta+1)/(beta-2) d. none of these

If the ratio of the roots of the equation x^2+p x+q=0 are equal to ratio of the roots of the equation x^2+b x+c=0 , then prove that p^2c=b^2qdot