Quadratic Equation and its Solution

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

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

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

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

Let $S=\left\{x:x\in R \mathrm{and} {\left(\sqrt{3}+\sqrt{2}\right)}^{x^2-4}+{\left(\sqrt{3}-\sqrt{2}\right)}^{x^2-4}=10\right\}$, then $n\left(S\right)$ is equal to

The number of real roots of the equation $\sqrt{x^2-4x+3}+\sqrt{x^2-9}=\sqrt{4x^2-14x+6}$, is:

Find the roots of the equation $a\left(x^2+1\right)-\left(a^2+1\right)x=0, \text{where }a\ne 0$

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

If $x+\frac{1}{x}=-2$, then find the value of $x^{96}-x^{95}+x^{86}-x^{84}+....+x^6-x^5$.

Solve the equation $\sqrt{a\left(2^x-2\right)+1}=1-2^x$ for every value of the parameter $a$.

Suppose $\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$, then the roots of $a{\left(x-2\right)}^2-b\left(x-2\right)\left(x-3\right)+c{\left(x-3\right)}^2=0$ are

If $x=\sqrt{4+3\sqrt{4+3\sqrt{4+\dots \dots }}}$ & $y=3+\frac{4}{3+\frac{4}{3+\dots \dots \infty }}$; Then -

If $x^2+xy=12$ and $2xy+3y^2+5=0$ then $x+4y$ is equal to:

If $a,b,c\in R$ such that $a+b+c=0$ and '$a$' is not equal to '$c$', then the roots of $\left(b+c-a\right)x^2+\left(c+a-b\right)x+\left(a+b-c\right)=0$ are

Let $\alpha$ and $\beta$ be the roots of the equation $x^2-5x-1=0$, then find the value of $\frac{{\alpha }^{15}+{\alpha }^{11}+{\beta }^{15}+{\beta }^{11}}{{\alpha }^{13}+{\beta }^{13}}$.

Find the number values of $x$ satisfying the equation $3x^2-4\left|x^2-1\right|+x-1=0$

Find the quadratic equation $y=a{x}^2+bx+c$ that passes through the points $\left(-2,20\right),\left(1,5\right)$ and $\left(3,25\right)$.