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For the equation $x^2+bx+c=0,$ if $1+b+c=0$ for all $b,c\in R,$ then the roots are

Find the value of b, if one root of the equation $x^2$ + ax + 2b = 0 is 2 + 4i, where a, b ∈ R

If $f\left(x\right)=x^2+2bx+2c^2$ and $g\left(x\right)=-x^2-2cx+b^2$ are such that the minimum value of $f\left(x\right)$ always exceeds maximum value of $g\left(x\right),$ then which of the following is/are correct?

The value of $a$ for which $x^3+ax+1=0$ and $x^4+a{x}^2+1=0$ have exactly one common root is

If $\alpha ,\beta ,\gamma ,\delta$ are the roots of equation $2x^4+4x^3-3x^2+3x+1=0$, then value of $\frac{1}{2\alpha -1}+\frac{1}{2\beta -1}+\frac{1}{2\gamma -1}+\frac{1}{2\delta -1}$ is

Plot the graph of $y=($x-3${)}^2+7$

If $\alpha ,\beta ,\gamma$ be the roots of the equation $(x-a)(x-b)(x-c)=d,d\ne 0$, then the roots of the equation $(x-\alpha )(x-\beta )(x-\gamma )+d=0$ are

If $a,b,c,d$ and $p$ are distinct non-zero real numbers such that $(a^2+b^2+c^2)p^2-2(ab+bc+dc)p+(b^2+c^2+d^2)\le 0$, then $ac$ is equal to

For the equation $|x{|}^2+|x|-6=0,\text{the sum of the real roots is}$

Let $p,q,r$ be roots of cubic equation $x^3+2x^2+3x+3=0$, then

The range of $a$ for which $x^2-ax+1-2a^2$ is always positive for all real values of $x$, is

The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation, $6x^2-11x+\alpha =0$ are rational numbers is :

If the zeroes of monic cubic polynomial are $3,5$ and $6,$ then the cubic polynomial is

If $x_1,x_2,x_3\cdots x_n$ are roots of $x^n+ax+b=0$, then the value of $(x_1-x_2)(x_1-x_3)(x_1-x_4)\cdots (x_1-x_n)$ =

The value(s) of $k$ for which the quadratic equations $(1-2k)x^2-6kx-1=0$ and $k{x}^2-x+1=0$ have at least one root in common, is (are)

If $x^2-ax+b=0$ and $x^2-px+q=0$ have one root common and the second equation has equal roots, then $b+q$ is equal to

The value of $y=2+\frac{1}{4+\frac{1}{4+\frac{1}{4+\frac{1}{4+......\infty }}}}$ is

Let $\alpha ,\beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha }{2},2\beta$ be the roots of the equation $x^2-qx+r=0$. Then, the value of $r$ is:

If $x^2+ax+bc=0,$ $x^2+bx+ac=0,$ $a\ne b$ have one root in common, then their other roots satisfy the equation

Solve the following quadratics $13x^2+7x+1=0$

If $x^2-2x+{\log }_{\frac{1}{2}}p=0$ does not have two distinct real roots, then the maximum value of $p$ is

If $p\left(x\right)$ is a polynomial of degree greater than $2$ such that $p\left(x\right)$ leaves remainder $a$ and $-a$ when divided by $x+a$ and $x-a$ respectively. If $p\left(x\right)$ is divided by $x^2-a^2$ then remainder is

Suppose $a^2=5a-8$ and $b^2=5b-8$. Then the equation whose roots are $\frac{a}{b}$ and $\frac{b}{a}$ is

If $\alpha ,\beta$ are the root of a quadratic equation $x^2-3x+5=0$, then the equation whose roots are $({\alpha }^2-3\alpha +7)\text{and}({\beta }^2-3\beta +7)$ is

If $1-p$ is a root of the equation $x^2+px+1-p=0$, then the roots of the equation are

If $k_1,k_2,k_3$ are real numbers and the equation $k_1{x}^2+k_{2}x+k_3=0$ have three roots, then value of $k_1+k_2+k_3$ is

Let $f\left(x\right)=x^5+a{x}^3+bx.$ The remainder when $f\left(x\right)$ is divided by $x+1$ is $-3.$ Then the remainder when $f\left(x\right)$ is divided by $x^2-1,$ is

Let p and q be real numbers such that p$\ne 0,p^3\ne q$ and $p^3\ne -q$ . If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha +\beta =-p$ and ${\alpha }^3+{\beta }^3=q$ , then a quadratic equation having $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ as its roots is -

Let $\alpha$ and $\beta$ are two real roots of the equation $(k+1){\tan }^{2}x-\sqrt{2}\lambda \tan x=1-k,$ where $k\ne -1$ and $\lambda$ are real numbers. If ${\tan }^2(\alpha +\beta )=50,$ then the value of $\lambda$ is

The number of real roots of the equation $(x^2+2x{)}^2-(x+1{)}^2-55=0$

The complete set of values of $a$ for which the inequality $a{x}^2-(3+2a)x+6>0,a\ne 0$ holds good for exactly three integral values of $x$ is

The set of values of $k$ for which the equation $(k+2)x^2-2kx-k=0$ has two roots on the number line symmetrically placed about $1$ is

If the product of the roots of the quadratic equation $m{x}^2-2x+(2m-1)=0$ is $3$, then the value of $m$ is

The values of $a$ for which the number $6$ lies in between the roots of the equation $x^2+2(a-3)x+9=0,$ belong to

Solve the following quadratics $17x^2-8x+1=0$