Roots of a Quadratic Equation

Q. The number of real roots of the equation $\frac{x^2-3x}{x^2-1}+2+\frac{1}{x-1}=0$ is

1. $1$
2. $0$
3. $2$
4. infinitely many

Q. If alpha, beta, gamma, delta are the roots of the equation x^4 + ax^3 + bx^2 + cx + d = 0, find the value of summation of alpha^2*beta

Q.

Find all the real fourth roots of $\frac{256}{2401}$.

Q.

The number of real values of x for which the equality $|3x^2+12x+6|=5x+16$ holds good is

1. 3

2. 1

3. 2

4. 4

Q.

If bi-quadratic equation $x^4-2x^3+4x^2+6x-21=0$ can also be written as $(x^2+p)(x^2-2x+q)=0$. Find the sum of p + q.

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Q. IF the roots of the equation x^2+2px+3q=0 differ by 1 , then (p^2-3q) =

Q.

How do you find the zeros of $y=x^4-7x^2+12$?

Q.

Find the roots of the following quadratic equation if they exist by the method of completing the square : $2x^2-7x+3=0$.

Q. Let $\alpha ,\beta$ be the roots of $x^2-x-1=0$ $(\alpha >\beta )$ and $m,n\in Z,k\in W$ such that $a_k=m{\alpha }^k+n{\beta }^k.$ If $a_4=35,$ then the value of $3m+2n$ is equal to

1. $40$
2. $85$
3. $25$
4. $75$

Q. sdsfd

1. sd
2. casc
3. csa
4. asc

Q.

The equation whose roots are twice the roots of the equation $x^2-3x+3=0$ is

1. $x^2-3x+6=0$

2. $x^2-4x+8=0$

3. $x^2-6x+12=0$

4. $x^2-8x+6=0$

Q. Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation, $x^2-x-1=0,$ where $\alpha \ne \beta .$ For $n=0,1,2,....,$ let $a_n=p{\alpha }^n+q{\beta }^n.$

FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0.$ then $a=0=b.$

$a_{12}=$

1. $a_{11}-a_{10}$
2. $a_{11}+a_{10}$
3. $2a_{11}+a_{10}$
4. $a_{11}+2a_{10}$

Q. If one root of the equation $x^2+(1-3i)x-2(1+i)=0$ is $-1+i,$ then the other root is

1. $-1-i$
2. $\frac{-1-i}{2}$
3. $i$
4. $2i$

Q.

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

1. $n{x}_1+b$

2. $n(x_1{)}^{n-1}$

3. $n(x_1{)}^{n-1}+a$

4. $(n{x}_1{)}^{n-1}+b$

Q.

Solve the quadratic equation $a(x^2+1)=x(a^2+1)$ by using the quadratic formula.

Q.

The roots of the equation $\sqrt{3x+1}+1$ = $\sqrt{x}$ are

1. roots doesnot exist

2. 0

3. 1

4. 0, 1

Q. The number of values of k. for which the system or equations:
(k+1) x +8y = 4k
kx + (k+3) y = 3k-1
has no solution, is

1. infinite
2. 1
3. 2
4. 3

Q. 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

1. $1$
2. $2$
3. $-1$
4. $3$

Q. Let $x^2-(m-3)x+m=0,m\in R$ be a quadratic equation. Then the set of value(s) of $m$ for which roots are real and distinct is/are

1. $(-\infty ,1)$
2. $(1,9)$
3. $(9,\infty )$
4. $\left\{1\right\}$

Q.

The roots of the equation $i{x}^2-4x-4i$ = 0 are

1. 2 , -2$i$

2. 2$i$

3. -2$i$, -2$i$

4. 2$i$, 2$i$

Q. If $\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$, then the quadratic equation whose roots are $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ is

1. $a{x}^2+bx+c=0$
2. $c{x}^2+ax+c=0$
3. $b{x}^2+ax+c=0$
4. $c{x}^2+bx+a=0$

Q. The roots of the equation $x^2+|x|-2=0$ is/are

1. $-1$
2. $-2$
3. $1$
4. $2$

Q.

The set of all real numbers x for which $x^2-|x+2|+x>0$, is

1. 

2. 

3. 

4. 

Q. The number of values of $x$ satisfying the equation $(x^2+7x+11{)}^{(x^2-4x-21)}=1$ is

1. $6$
2. $4$
3. $2$
4. $5$

Q. Let $p,q$ be integers and $\alpha ,\beta$ be the roots of the equation $x^2-2x+3=0$ where $\alpha \ne \beta$.
$\text{If}{a}_n=p{\alpha }^n+q{\beta }^n\text{where}n\in \{0,1,2,.....\}$, then the value of $a_9$ is

1. $3a_8-5a_7+3a_6$
2. $a_7-6a_6$
3. $a_7-12a_5+18a_4$
4. $2a_8+3a_6$

Q. The value of
$x=1+\frac{1}{3+\frac{1}{2+\frac{1}{3+\frac{1}{2+.....\infty }}}}$ is

1. $\sqrt{\frac{5}{3}}$
2. $-\sqrt{\frac{5}{3}}$
3. $\sqrt{\frac{3}{2}}$
4. $-\sqrt{\frac{3}{2}}$

Q.

The number of real solutions of the equation $|x{|}^2-3|x|+2=0$ are

1. 1

2. 2

3. 3

4. 4

Q.

If two roots of biquadratic equation $x^4+12x-5=0$ are -1 + $\sqrt{2}and1+\sqrt{2}i$ . Find the sum of the square of all the roots.

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Q. $27x^2-10+1=0$

Q.

The number of real values of x for which the equality $|3x^2+12x+6|=5x+16$ holds good is

1. 4

2. 3

3. 2

4. 1