Quadratic Equations

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

Solution to the trigonometric equation ${\sin }^{2}x – 5\sin x + 4 = 0$ is 

Formulate into a mathematical problem to find a number such that when its cube root is added to it, the result is $6$.

Find all real numbers satisfying $4^x-3\left(2^{x+2}\right)+2^5=0$.

If $\alpha ,\beta$ are the roots of the equation $x^2-7x-4=0$, then ${\alpha }^3=53\alpha +k$. Find the value of $k$?

Solve the equation $\sqrt{3x+4}=x$.

A trader bought a number of articles for $₹1200$. Ten were damaged and he sold each of the rest at$₹2$ more than what he paid for, thus clearing a profit of $₹60$ on the whole transaction. Find the number of articles he bought.

A piece of cloth costs $₹175$. If the piece were $4 \mathrm{m}$ longer and each metre costs $₹5$ less, the cost would have remains unchanged. How long is the piece?

 The roots of the equation $\frac{x+3}{x-2}-\frac{1-x}{x}=\frac{17}{4}$ are

Subtract the quadratic expression $9x^2-11x+5 \mathrm{from} 3x^2+x-1.$

If $6x^6-25x^5+31x^4-31x^2+25x-6=0$, then sum of all the roots of the equation is

If $1, 1, \alpha$ are the roots of $x^3-6x^2+9x-4=0$ then find $\alpha$.

If $\alpha , \beta , \gamma$ are the roots of $4x^3-6x^2+7x+3=0$ then find $a\beta +\beta \gamma +\gamma \alpha$

The number of cubic polynomials $P\left(x\right)$ satisfying $P\left(1\right)=2, P\left(2\right)=4, P\left(3\right)=6, P\left(4\right)=8$ is
 

The number of real roots of the polynomial equation $x^4-x^2+2x-1=0$ is

If $1, 1, \alpha$ are the roots of the equation $x^3-6x^2+9x-4$, then the value of $\alpha$ is

Find the sum of zeros of the polynomials $x^2+6x+9=0$.

In a class, the total number of boys and girls are in the ratio $4: 3$. On one day it was found that $8$ boys and $14$ girls were absent from the class and that the number of boys was the square of the number of girls. What is the sum of digits of total number of students in the class?

In a class, the total numbers of boys and girls are in the ratio $4: 3$. On one day it was found that $8$ boys and $14$ girls were absent from the class and that the number of boys was the square of the number of girls. What is the sum of digits of total number of students in the class?

In the question, two equations I and II with variables $\mathrm{a}$ and $\mathrm{b}$ are given. You have to solve both the equations to find the values of $\mathrm{a}$ and $\mathrm{b}$.

I. ${\mathrm{a}}^2 – 4\sqrt{3\mathrm{a}} + 9 = 0$

II. ${\mathrm{b}}^2 – 5\sqrt{3\mathrm{b}} + 18 = 0$