Multiple choice If x_1, x_2 are two real roots of the quadratic equation x^2-mx+m-2=0, then what statement is correct about m to hold the equation \dfrac{1}{x_1}+\dfrac{1}{x_2}=0 true?

Q&A If a,b,c are real numbers such that a+b+c=2, \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{2}, show that one of numbers among a,b,c must be equal to 2

Q&A If a,b,c are real numbers such than a=2b+\sqrt{2} and ab+\dfrac{\sqrt{3}}{ 2}c^2+\dfrac{1}{4}=0 , find the value of \dfrac{bc}{a}

Q&A Let p(x) be a cubic polynomial with integer coefficients with leading coefficient 1 and with one of its roots equal to the product of the other two. Show that 2p(−1) is a multiple of p(1) + p(−1) − 2(1 + p(0))

Q&A The product of two of the four roots of the quartic equation x^4 − 18x^3 + kx^2 + 200x − 1984 = 0 is −32. Determine the value of k.

Q&A If x_1 and x_2 are the two roots of the quadratic equation x^2-px+q=0, (x_1>x_2). Find the values of the following expressions.

Q&A Let m and n be the two roots of the equation x^2 − 15x + 28 = 0. Find the value of (m + 1)(n + 1)

Q&A Let x_1 and x_2 be the roots of the equation x^2 + 3x + 1 = 0. Compute\Big( \dfrac{x_1}{x_2+1} \Big) ^2+\Big( \dfrac{x_2}{x_1+1} \Big) ^2

Q&A Let x_1 and x_2 be the roots of the equation x^2 − (a + d)x + (ad − bc) = 0. Show that x^3_1 and x^3_2 are the roots of the equationy^2 − (a^3 + d^3 + 3abc + 3bcd)y + (ad − bc)^3 = 0

Q&A The depressed cubic equation x^3 +px + 28=0 has three distinct roots. Two of these roots sum to 2. Find the the value of p.