Quadratic Equations with Both Roots Common

Q. If the equations \(4x^2 -x-1 = 0\) and \(3x^2 +(\lambda + \mu)x + \lambda - \mu = 0\) have a common root, then the rational values of \(\lambda\) and \(\mu\) are

Q. If equation $(k-1)x^2+(k^2+1)x+6=0$ and $2x^2+10x+12=0$ have both roots common, then the value of $k$ is ___

Q. The two equations $x^3+1=0$ and $a{x}^2+bx+c=0,a,b,c\in R$ have two roots in common. Then $a+b$ is equal to

1. $-1$
2. $2$
3. $0$
4. $3$

Q. If $a,b,c$ be the sides of $△ABC$ and equations $a{x}^2+bx+c=0$ and $5x^2+12x+13=0$ have a common root, then $\angle C$ is

1. ${90}^{\circ }$
2. ${45}^{\circ }$
3. ${120}^{\circ }$
4. ${60}^{\circ }$

Q. If $p{x}^2+5x+2=0$ and $3x^2+10x+q=0$ have both roots in common, then

1. $q=4$
2. $p=\frac{3}{2}$
3. $p=4$
4. $q=\frac{3}{2}$

Q.

If $a,b,c\in R$ and equations $a{x}^2+bx+c=0$ and $2x^2+4x+6=0$ have a common root with $a+b+c=18$, the value of $a^{2}bc$ is

__

Q. If equations $x^2-3x+4=0$ and $4x^2-2[3a+b]x+b=0(a,b\in R)$ have a common root, then the complete set of values of $a$ is
$($Here, $\left[K\right]$ denotes the greatest integer less than or equal to $K.)$

1. $[\frac{-10}{3},-3)$
2. $(3,\frac{10}{3}]$
3. $(\frac{-10}{3},-3]$
4. $[\frac{-11}{3},\frac{-10}{3})$

Q.

If the equations $a{x}^2+bx+1=0$ and $2x^2+4x+3=0$ have both the roots common, then the value of $a+b$ is:

1. $4$

2. $1$

3. $2$

4. $3$

Q.

If $x^2+7x+1=0$ and $(a^2-4a)x+bx-3=0$ have both the roots in common, then find the value of $a$.

1. $2$

2. $3$

3. $4$

4. $1$

Q.

If the equations $x^2+2x+3=0$ and $a{x}^2+bx+c=0$ , $a,b,c\in R$ , have a common root, then $a:b:c$ is

1. $1:2:3$

2. $3:1:2$

3. $3:2:1$

4. $1:3:2$

Q. Choose the correct pair of quadratic equations with both roots common.

1. $2x^2-6x-12=0$
2. $4x^2+2x=0$
3. $2x^2+x=0$
4. $x^2+2x+2=0$
5. $-x^2+3x+6=0$
6. $2x^2+4x+4=0$

Q.

If $a,b,c\in R$ and the equations a$x^2+bx+c=0$ and $x^3+3x^2+3x+2=0$ have two roots in common, then

1. $a=b=c$

2. $a\ne b\ne c$

3. $a=b\ne c$

4. $a=b\ne -c$

Q. For two quadratic equations to have both roots common, the ratio of the corresponding coefficients must be equal.

1. False
2. True

Q. The equations $a{x}^2+bx+c=0$ and $x^2+x+c=0$ have
( where $c\ge 1$ )

1. a common root only when $a=b\ne c$ and $a,b\in R$
2. two common roots only when $a=b=c$
3. two common roots when $a,b\in R$
4. a common root when $a,b\in R$

Q. Find the value of $m$ for which the equations $x^2+mx+1=0\&(b-c)x^2+(c-a)x+(a-b)=0$ have both roots common.

1. $-2$
2. $1$
3. $2$
4. $0$