If the sum of the roots of the quadratic equation ax²+ bx + c = 0 (abc ≠ 0)  is equal to the sum of the squares of their reciprocals, then \(\frac { a }{ c } ,\frac { b }{ a } ,\frac { c }{ b } \)  are H.P.

If a, b, c, d and p are distinct non-zero real numbers such that (a²+b²+c²) p²-2 (ab+bc+cd) p+(b²+c²+d²)≤ 0 then prove that a, b, c, d are in G.P and ad = bc

If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p. 

If the equation x² + bx + ca = 0 and x² + cx + ab = 0 have a comnion root and b≠c, then prove that their roots will satisfy the equation x² + ax + bc = 0.

Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

Solve the equation \(3 x^{3}-x+88=4 x^{2}\) if one of the root of the given cubic polynomial equation is \(2-\sqrt{7} i\)

Solve \(4 x^{5}+x^{3}+x^{2}-3 x+1=0\), given that it has rational roots.

Show that the equation \(x^{3}+q x+r=0\) has two equal roots if \(27 r^{2}+4 q^{3}=0\).

Show that the roots of the equation \(a x^{3}+b x^{2}+c x+d=0\) are in geometric progression, then \(c^{3} a=b^{3} d\)

If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^{3}+p x^{2}+q x+r=0\), find the value of the following in terms of the co-efficients.
\((i)\ \sum \frac{1}{\beta \gamma} \)
\((ii)\ \sum \frac{1}{\alpha} \)
\((iii)\ \sum \alpha^{2} \beta \)

lf \(\alpha\) is an imaginary root of the equation \(x^{7}-1=0\) form the equation whose roots are \(\alpha+\alpha^{6}, \alpha^{2}+\alpha^{5}, \alpha^{3}+\alpha^{4}\)

 If \(\alpha, \beta, y\) are the roots of \(x^{3}+3 x^{2}+2 x+1=0\), find \(\sum \alpha^{3} \text { and } \sum \alpha^{-2}\)

lf \(\alpha, \beta, \gamma\) are the roots of \(x^{3}-x+1=0\), show \(\frac{1+\alpha}{1-\alpha}+\frac{1+\beta}{1-\beta}+\frac{1+\gamma}{1-\gamma}=1\)

Solve the equation
\(60 x^{4}-736 x^{3}+1433 x^{2}-736 x+60=0\)

Solve: \(x^{5}-5 x^{4}+9 x^{3}-9 x^{2}+5 x-1=0\)