Amit M Agarwal Solutions for Chapter: Inequalities and Quadratic Equation, Exercise 6: Target Exercises

Suppose $A, B$ and $C$ are defined as $A=a^{2}b+a{b}^2-a^{2}c-a{c}^2, B=b^{2}c+b{c}^2-a^{2}b-a{b}^2$ and $C=a^{2}c+a{c}^2-b^{2}c-b{c}^2$, where $a>b>c>0$ and the equation $A{x}^2+Bx+C=0$ has equal roots, then $a, b$ and $c$ are in

If $\alpha$ and $\beta$ are the roots of $a{x}^2+c=bx$, then the equation ${\left(a+cy\right)}^2=b^{2}y$ in $y$ has the roots

The equation formed by decreasing each root of $a{x}^2+bx+c=0$ by $1$ is $2x^2+8x+2=0,$ then

If the equations $a{x}^2+bx+c=0$ and $x^2+x+1=0$ have a common roots, then

If the equations $a{x}^2+bx+c=0$ and $c{x}^2+bx+a=0, a\ne c$ have a negative common root, then the value of $a-b+c$ is

If the equations $x^2+ix+a=0$ and $x^2-2x+ia=0$ $a\ne 0$ have a common root, then $a$ is equal to 

If the equations $x^2+2x+3\lambda =0$ and $2x^2+3x+5\lambda =0$ have a non-zero common root, then $\lambda$ is equal to

If the equations $a{x}^2+bx+c=0$ and $x^2+2x+3=0$ have a common root, then $a:b:c$ is equal to