[" Show that the "700" ts "sigma_(f)(x-a)(-b)=h^(2)],[" are alcoays real? "]

Prove that the roots of (x-a)(x-b)=h^(2) are always real.

If a,b, are real, then show that the roots of the equation (x-a)(x-b) = b^2 are always real.

If a, b are real then show that the roots of the equation (a-b)x^(2)-6(a+b)x-9(a-b)=0 are real and unequal.

If a and b are real and aneb then show that the roots of the equation (a-b)x^(2)+5(a+b)x-2(a-b)=0 are real and unequal.

If x is real , show that the expressions (x^2-ab)/(2x-a-b) has no real value between a and b.

Show that the function f (x)=x^(2)+2x is continuous for every real value of x .

Given a differentiable function f(x) defined for all real x, and is such that f(x+h)-f(x)<=6h^(2) for all real h and x . Show that f(x) is constant.

If f(x) is a real valued polynomial and f (x) = 0 has real and distinct roots, show that the (f'(x)^(2)-f(x)f''(x))=0 can not have real roots.

If f(x) is a real valued polynomial and f (x) = 0 has real and distinct roots, show that the (f'(x)^(2)-f(x)f''(x))=0 can not have real roots.

IF a,b,c are real, show that the roots of each of the following equations are real: (x-a)(x-b)=b^2