Quadratic Equation with Complex Roots

Find the value of $k$, if $x-1$ is a factor of $4x^3+3x^2-4x+k$.

$\left(1+m^2\right)x^2+2 mcx+c^2-a^2=0$ has equal roots if ________.

What is the value of $\text{k}$ that $(2\mathrm{x}-1)$ may be a factor of $4{\mathrm{x}}^4-(\mathrm{k}-1){\mathrm{x}}^3+{\mathrm{kx}}^2-6\mathrm{x}+1$ ?

If $p$ and $q$ are the roots of the polynomial $p\left(x\right)=a{x}^2+bx+c$,the polynomial $g\left(x\right)$ whose roots are $\frac{1}{p}$ and $\frac{1}{q}$ will be?

What is the solution of the equation $\sqrt{\frac{x}{x+3}}-\sqrt{\frac{x+3}{x}}=-\frac{3}{2}$?

Which of the following equations has real roots$?$

The roots of equation $y^3+y^2-y-1=0$ are

If one root of the quadratic equation $a{x}^2+bx+c=0$ is equal to nth power of the other root, then the value of ${\left(a{c}^n\right)}^{\frac{1}{n+1}}+{\left(a^{n}c\right)}^{\frac{1}{n+1}}$ is equal to

The system of linear equations $x+2y+z=4, 2 x+4y+2z=8$ and $3x+6y+3z=10$ has

$\alpha$ and $\beta$ are distinct real roots of the quadratic equation $x^2+ax+b=0$. Which of the following statements is/are sufficient to find $\alpha$?

$1.$ $\alpha +\beta =0, {\alpha }^2+{\beta }^2=2$

$2.$ $\alpha {\beta }^2=-1, a=0$

Select the correct answer using the code given below :

Solve the following quadratic equation $2x^2+3ix+2=0$

If $a$ and $b$ are the roots of the equation $x^2-7x-1=0$, then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to

For what values of $m,$ is $m{x}^2+mx+8x+9$ a perfect square?

Consider the following statements in respect of the polynomial $1-x-x^n+x^{n+1},$ where $n$ is a natural number:

It is divisible by $1-2x+x^2.$

It is divisible by $1-x^n.$

Which of the statements given above is/are correct?

Consider the following statement in respect of the polynomial $a(b-c)(x-b)(x-c)+b(c-a)(x-c)(x-a)+c(a-b)(x-a)(x-b):$

1. The coefficient of $x^2$ is $0.$

2. The coefficient of $x$ is $(a-b)(b-c)(c-a).$

Which of the statements given above is/are correct?

The number of real roots of the equation $x\left|x\right|-5\left|x+2\right|+6=0$, is

The number of solution equation $x\left|x\right|-5\left|x+2\right|+6=0$ is:

Let $a$ and $b$ are roots of $x^2-7x-1=0$, then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is

The sum of roots of $\left|x^2-8x+15\right|-2x+7=0$ is

Solve the following quadratic equation $2x^2+3ix+2=0$