Home > Condition for Common Roots

Condition for Common Roots - (Concept)

We consider two-equation and try to find the conditions for roots to be common among them

Let the equations be $\mathrm{a}_1 \mathrm{x}^2+\mathrm{b}_1 \mathrm{x}+\mathrm{c}_1=0$ and $\mathrm{a}_2 \mathrm{x}^2+\mathrm{b}_2 \mathrm{x}+\mathrm{c}_2=0$

Only one common root:

Let ? be the common root, so it will satisfy both the equations

This, $\mathrm{a}_1 \alpha^2+\mathrm{b}_1 \alpha+\mathrm{c}_1=0$ and $\mathrm{a}_2 \alpha^2+\mathrm{b}_2 \alpha+\mathrm{c}_2=0$

In solving these two equations using the multiplication method, we get

$
\begin{aligned}
& \frac{\alpha^2}{\mathrm{~b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1}=\frac{\alpha}{\mathrm{c}_1 \mathrm{a}_2-\mathrm{c}_2 \mathrm{a}_1}=\frac{1}{\mathrm{a}_1 \mathrm{~b}_2-\mathrm{a}_2 \mathrm{~b}_1} \\
& \Rightarrow \alpha^2=\frac{\mathrm{b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1}{\mathrm{a}_1 \mathrm{~b}_2-\mathrm{a}_2 \mathrm{~b}_1}
\end{aligned}
$

and,

$
\Rightarrow \alpha=\frac{\mathrm{c}_1 \mathrm{a}_2-\mathrm{c}_2 \mathrm{a}_1}{\mathrm{a}_1 \mathrm{~b}_2-\mathrm{a}_2 \mathrm{~b}_1}
$

from the above equations, we can write

$
\begin{aligned}
& \Rightarrow \frac{\mathrm{b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1}{\mathrm{a}_1 \mathrm{~b}_2-\mathrm{a}_2 \mathrm{~b}_1}=\left(\frac{\mathrm{c}_1 \mathrm{a}_2-\mathrm{c}_2 \mathrm{a}_1}{\mathrm{a}_1 \mathrm{~b}_2-\mathrm{a}_2 \mathrm{~b}_1}\right)^2 \\
& \Rightarrow\left(\mathbf{b}_{\mathbf{1}} \mathbf{c}_{\mathbf{2}}-\mathbf{b}_{\mathbf{2}} \mathbf{c}_{\mathbf{1}}\right)\left(\mathbf{a}_{\mathbf{1}} \mathbf{b}_{\mathbf{2}}-\mathbf{a}_{\mathbf{2}} \mathbf{b}_{\mathbf{1}}\right)=\left(\mathbf{c}_{\mathbf{1}} \mathbf{a}_{\mathbf{2}}-\mathbf{c}_{\mathbf{2}} \mathbf{a}_{\mathbf{1}}\right)^{\mathbf{2}}
\end{aligned}
$

This is the condition required for a root to be common to both quadratic equations.

The common root is given by,

$
\alpha=\frac{b_1 c_2-b_2 c_1}{a_1 b_2-a_2 b_1}=\frac{c_1 a_2-c_2 a_1}{a_1 b_2-a_2 b_1}
$

The common root can also be found using the method given below :

First, make the coefficient of x^2the same in two given quadratic equations.
Now, subtract the two equations
Use the relation between their roots and coefficients to find the roots of other equations.

Both roots are common :

Let ? and ? be the common roots of the equations

$
\mathrm{a}_1 \mathrm{x}^2+\mathrm{b}_1 \mathrm{x}+\mathrm{c}_1=0 \text { and } \mathrm{a}_2 \mathrm{x}^2+\mathrm{b}_2 \mathrm{x}+\mathrm{c}_2=0
$

Then, both the equation are identical, hence

$
\frac{\mathrm{a}_1}{\mathrm{a}_2}=\frac{\mathrm{b}_1}{\mathrm{~b}_2}=\frac{\mathrm{c}_1}{\mathrm{c}_2}
$