Common Roots

If $a, b, c\in R$, then find the relation between $a, b, c$ so that the two quadratic equations $a{x}^2+bx+c=0$ and $1003x^2+1505x+2007=0$ have a common root.

If equations $x^6+a{x}^4-4x=0$ and $x^7+a{x}^5-1=0$ have a common root, then identify which of the following statement is(are) correct?

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 $a{x}^2+bx+c=0$ and $c{x}^2+bx+a=0$, where $a\ne c$ have a common root, then $a+b+c=0$ or $a-b+c=0$

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 $k$.

If one root of the equations $a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0\left(a, b, c\in R\right)$ is common, then find the value of ${\left(\frac{a^3+b^3+c^3}{abc}\right)}^3$

If $x^2+3x+5=0$ and $a{x}^2+bx+c=0$ have common root / roots and $a, b, c\in N.$ then the minimum value of $a+b+c$ is 

If all the equations ${\mathrm{x}}^2+(2\mathrm{a}+3\mathrm{b})\mathrm{x}+60=0, {\mathrm{x}}^2+\mathrm{ax}+10=0$ and ${\mathrm{x}}^2+\mathrm{bx}+8=0$ where $\mathrm{a}, \mathrm{b}\in \mathrm{R},$ have a common root, then value of $|\mathrm{a}-\mathrm{b}|$ is

If the equation $x^2+px+2q=0$ and $x^2+qx+2p=0(\mathrm{p}\ne \mathrm{q})$ have a common root
then the absolute value of $(p+q)$ is

If all the equations $x^2+(2a+3b)x+60=0,x^2+ax+10=0$ and $x^2+bx+8=0$ where $a,b\in R,$have a common root, then value of $|\mathrm{a}-\mathrm{b}|$ is

If $b{x}^2+ax+c=0, a{x}^2+bx+c=0$ and $a{x}^2+cx+b=0,$ each equation has equal roots, then $\left|\begin{array}{lll}3& a& bc\\ 3& b& ac\\ 3& c& ab\end{array}\right|$ is equal to

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 ?