Common Roots of Quadratic Equations

In this section, we shall obtain conditions under which two given quadratic equations may have common roots.

One Root Common

Let two quadratic equations be \[\begin{array}{l} & ax^2+bx+c &= 0 \\ \text{and,} & a’x^2+b’x+c’ &= 0 \end{array}\]

Let $\alpha$ be a root common to both the equations. Then, \[\begin{array}{l} & a\alpha^2+b\alpha+c &= 0 \\ \text{and,} & a’\alpha^2+b’\alpha+c &= 0 \end{array}\]

By the rule of cross multiplication, \[\frac{\alpha^2}{bc’-b’c}=\frac{\alpha}{ca’-c’a}=\frac{1}{ab’-a’b}\] \[\therefore\alpha=\frac{bc’-b’c}{ca’-c’a}=\frac{ca’-c’a}{ab’-a’b}\] \[\therefore\frac{bc’-b’c}{ca’-c’a}=\frac{ca’-c’a}{ab’-a’b}\] \[\text{or,}\;\;\;(bc’-b’c)(ab’-a’b)=(ca’-c’a)^2\] This is the required condition and the common root is \[\frac{bc’-b’c}{ca’-c’a}\;\;\;\text{or,}\;\;\;\frac{ca’-c’a}{ab’-a’b}\]

Both Roots Common

Let $\alpha$ and $\beta$ be the common roots of the quadratic equations \[\begin{array}{l} & ax^2+bx+c &= 0 \\ \text{and,} & a’x^2+b’x+c’ &= 0 \end{array}\]

Then, \[\alpha+\beta=-\frac{b}{a}=-\frac{b’}{a’}\] \[\therefore\frac{a}{a’}=\frac{b}{b’}\] [From: Roots and Coefficients of a Quadratic Equation]

Again, \[\alpha\beta=\frac{c}{a}=\frac{c’}{a’}\] \[\therefore\frac{a}{a’}=\frac{c}{c’}\]

Hence, \[\frac{a}{a’}=\frac{b}{b’}=\frac{c}{c’}\] This is the required condition for the quadratic equations to have both roots common.


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