Quadratic Equations

If $\alpha$ is a common root of $x^2-5 x+\lambda=0$ and $x^2-8 x-2 \lambda=0(\lambda \neq 0)$ and $\beta, \gamma$ are the other roots of them, then $\alpha+\beta+\gamma+\lambda=$

If the sum of two roots of $x^3+p x^2+q x-5=0$ is equal to its third root, then $p\left(p^2-4 q\right)=$

If $\alpha$ and $\beta$ are two double roots of $x^2+3(a+3) x-9 a=0$ for different values of $a(\alpha>\beta)$, then the minimum value of $x^2+\alpha x-\beta=0$ is

If $2 x^2+3 x-2=0$ and $3 x^2+\alpha x-2=0$ have one common root, then the sum of all possible values of $\alpha$ is

If the roots of $\sqrt{\frac{1-y}{y}}+\sqrt{\frac{y}{1-y}}=\frac{5}{2}$ are $\alpha$ and $\beta(\beta>\alpha)$ and the equation $(\alpha+\beta) x^4-25 \alpha \beta x^2+(\gamma+\beta-\alpha)=0$ has real roots, then a possible value of $\gamm...

For any real value of $x$. If $\frac{11 x^2+12 x+6}{x^2+4 x+2} \notin(a, b)$, then the value $x$ for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is

Let $[r]$ denote the largest integer not exceeditio $r$ and the roots of the equation $3 x^2+6 x+5+\alpha\left(x^2+2 x+2\right)=0$ are complex number when ever $\alpha>L$ and $\alpha

If $\alpha, \beta, \gamma$ are roots of equations $x^3+a x^2+b x+x=0$, then $\alpha^{-1}+\beta^{-1}+\gamma^{-1}=$

If $x^2+5 a x+6=0$ and $x^2+3 a x+2=0$ have a common root, then that common root is

\(4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \infty}}}=\)

If the sum of two roots $\alpha, \beta$ of the equation $x^4-x^3-8 x^2+2 x+12=0$ is zero and $\gamma, \delta(\gamma>\delta)$ are its other roots, then $3 \gamma+2 \delta=$

If the roots of the quadratic equation $ x^2 - 35x + c = 0 $ are in the ratio 2 : 3 and $ c = 6K $, then $ K = $

For all positive integers $ n $ if $ 3^{2n+1} + 2^{n+1} $ is divisible by $ k $, then the number of prime numbers less than or equal to $ k $ is

If -1 is a twice repeated root of the equation $a\left(x^3+x^2\right)+b x+c=0$, then $a: b: c=$

If the values of $k$ for which the euqation $x^2+2(k+2) x+6 k+7=0$ has equal roots are $k_1$ and $k_2$, then $k_1^2+k_2^2=$

If the equation $x^4+a x^3+b x^2+c x+d=0$ has three equal roots, then that root is

If $(3+2 \sqrt{2})^{x^2-4}+(3-2 \sqrt{2})^{x^2-4}=6$, then $x^4+x^2+5=$

The minimum value of $f(x)=\frac{x^2-2 x+3}{x^2-4 x+7}$ is

If -1 is a twice repeated root of the equation $a x^3+b x^2+c x+1=0$, then

$\alpha$ and $\beta$ are the roots of the equation $x^2-a x+b=0$. If $\alpha^2+\beta^2$ and $\alpha^3+\beta^3$ are the roots of the equation $A x^2+B x+C=0$, then $C=$

If one root of the equation $a x^3+b x+c=0$ is twice another root, then

If one root of the cubic equation \(x^3+36=7 x^2\) is double of another, then the number of negative roots are

The sum of the real roots of the equation \(x^4-2 x^3+x-380=0\) is

If \(S=\left\{m \in R: x^2-2(1+3 m) x+7(3+2 m)=0\right.\) has distinct roots}, then the number of elements in \(S\) is

The number of real roots of the equation \(\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}\) is

If \(f(x)=a x^2+b x+c\) for some \(a, b, c \in R\) with \(a+b+c=3\) and \(f(x+y)=f(x)+f(y)+x y, \forall x, y \in R\). Then, \(\sum_\limits{n=1}^{10} f(n)=\)

The number of positive real roots of the equation \(3^{x+1}+3^{-x+1}=10\) is

For what values of \(a \in Z\), the quadratic expression \((x+a)(x+1991)+1\) can be factorised as \((x+b)(x+c)\), where \(b, c \in Z\) ?

If the difference between the roots of \(x^2+a x+b=0\) and that of the roots of \(x^2+b x+a=0\) is same and \(a \neq b\), then

If \(\frac{13 x+43}{2 x^2+17 x+30}=\frac{A}{2 x+5}+\frac{B}{x+6}\), then \(A^2+B^2=\)

If \(f(f(0))=0\), where \(f(x)=x^2+a x+b, b \neq 0\), then \(a+b=\)

The sum of the real roots of the equation \(|x-2|^2+|x-2|-2=0\) is

If \(f(x)=2x^3+mx^2-13x+n\) and 2, 3 are the roots of the equation \(f(x)=0\), then the values of m and n are

If \(2, 1\) and \(1\) are roots of the equation \(x^3-4 x^2+5 x-2=0\), then the roots of \(\left(x+\frac{1}{3}\right)^3-4\left(x+\frac{1}{3}\right)^2+5\left(x+\frac{1}{3}\right)-2=0\)

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2+x+1=0\), then the equation whose roots are \(\alpha^{2021}, \beta^{2021}\) is given by

\(2+\sqrt{5}, 1\) are roots of the cubic equation given by

If the product of the roots of \(9x^3+112x^2-120x+a=0\) is 12, then the value of \(a\) is

For \(a\ne b\), if the equation \(x^2+ax+b=0\) and \(x^2+bx+a=0\) have a common root, then the value of \(a+b\) is equal to

The sum of the roots of the equation \(e^{4 t}-10 e^{3 t}+29 e^{2 t}-22 e^t+4=0\) is

If \(a\) is a positive integer such that roots of the equation \(7 x^2-13 x+a=0\) are rational numbers, then the smallest possible value of \(a\) is

The value of \(a\) for which the equations \(x^3+a x+1=0\) and \(x^4+a x^2+1=0\) have a common root is

If \(\alpha\) and \(\beta\) are the roots of \(11 x^2+12 x-13=0\), then \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}\) is equal to (approximately close to)