Common Roots

If the equations $x^2+ax+1=0$ and $x^2-x-a=0$ have a real common root $b$$,$ then the value of $b$ is equal to

Let $f\left(x\right)=x^3-3x+b$ and $g\left(x\right)=x^2+bx-3,$ where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f\left(x\right)=0$ and $g\left(x\right)=0$ have a common root ?

If the equations $x^2+px+q=0$ and $x^2+rx+s=0$ have a common root, show that it must be $\frac{ps-rq}{q-s}$ or $\frac{q-s}{r-p}$.

The equations $x^2-cx+d=0$ and $x^2-ax+b=0$ have one common root and the $2\mathrm{nd}$ equation has equal roots. Prove that $2\left(b+d\right)=ac$.

Find the value of $k$ for which the equations $x^2-kx-21=0$ and $x^2-3kx+35=0$ have a common root.

If the equation $a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0$ (where $ac\ne b^2$) have a common root, then show that, either $a+b+c\mathit{=}0$ or $a=b=c$.

Show that the equations $\left(b-c\right)x^2+\left(c-a\right)x+\left(a-b\right)=0$ and $\left(c-a\right)x^2+\left(a-b\right)x+\left(b-c\right)=0$ have a common root.

If the quadratic equations $x^2+ax+b=0$ and $x^2+bx+a=0 (a\ne b)$ have a common root then find $\left(a+b\right)$.

If the equations $x^2+bx+ca=0$ and $x^2+cx+ab=0$ have exactly one non-zero common root, then prove that the other roots of the equations satisfy $x^2+ax+bc=0$.

If $\alpha , \beta$ are the roots of the equation $x^2+px+q=0$ and $\gamma , \delta$  are the roots of the equation $x^2+rx+s=0$ evaluate $(\alpha -\gamma )(\alpha -\delta )(\beta -\gamma )(\beta -\delta )$ in term of $p, q, r, s$ Hence, show that $(s-q{)}^2=(r-p\left)\right(ps-rq)$ is the condition for the existence of a common root of the two equations.

If the quadratic equations $x^2+ax+b=0$ and $x^2+bx+a=0(a\ne b)$ have a common root then find $(a+b)$.

If the equations $x^2-11x+k=0$ and $x^2-14x+2k=0$ have a common root, find the sum of possible values of $k$.

For next two question please follow the same  

 If the quadratic equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 have exactly one common root, then the relation between their coefficients is ${\left({\text{c}}_1{\text{a}}_2-{\text{c}}_2{\text{a}}_1\right)}^2=\left({\text{b}}_1{\text{c}}_2-{\text{b}}_2{\text{c}}_1\right)\left({\text{a}}_1{\text{b}}_2-{\text{a}}_2{\text{b}}_1\right)$. If both the roots are common, then the relation between their coefficient is  $\frac{{\text{a}}_1}{{\text{a}}_2}=\frac{{\text{b}}_1}{{\text{b}}_2}=\frac{{\text{c}}_1}{{\text{c}}_2}$.

 If the equations $a{x}^3+3b{x}^2+3cx+d=0$ and  $a{x}^2+2bx+c=0$  have a common root, then which of the following option is correct ?

If the quadratic equations $a{x}^2+2bx+3c=0$ and $3x^2+8x+15=0$ have a common root (where $a,b,c$ are the lengths of sides of a $∆ABC$), then $\left({\sin }^{2}A +{\sin }^{2}B+{\sin }^{2}C\right)=$_____ 

If $a, b, c$ are in A.P. and if $\left(b-c\right)x^2+\left(c-a\right)x+\left(a-b\right)=0$ and $2\left(c+a\right)x^2+\left(b+c\right)x=0$ have common root then which one is correct

If equations $a{x}^2+bx+c=0; a, b, c\in R \& c\ne 0$ and $2x^2+3x+4=0$ have a common root, then $a:b:c$ equals 

If the quadratic equations $k\left(6x^2+3\right)+rx+2x^2-1=0$ and  $6k\left(2x^2+1\right)+px+4x^2-2=0$ have both the roots common, then  $2r-p$ is equal to

If the equations $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root and $a, b$ and $c$ are in geometric progression, then $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in:

If the equations $x^4+ax=1$ and $x^5+a{x}^2+1=0$ have a common root, then

If the equations $2x^2-7x+9=0$ and ${ax}^2+bx+18=0$ have a common root, then $\left(a,\mathrm{}b\in R\right)$