QUADRATIC EQUATIONS - COMMON ROOTS OF TWO QUADRATIC EQUATIONS

QUADRATIC EQUATIONS

a. Only one common root

Let $\alpha$ be the common root of $a{x}^{2}+bx+c\, =0\, \& a'{x}^{2}+b'x+c'=0\,$ then

$a{\alpha }^{2}+b\alpha +c\, =0\, \& a'{\alpha }^{2}+b'\alpha +c'=0\,$ By Cramer's Rule

$\frac {{\alpha }^{2}} {bc'-b'c}=\frac {\alpha } {a'c-ac'}=\frac {1} {ab'-a'b}$

Therefore ,

$\alpha \, =\, \frac {ca'-c'a} {ab'-a'b}=\frac {bc'-b'c} {a'c-ac'}$

So the condition for a common root is

${(ca'-c'a)}^{2}=(ab'-a'b)(bc'-b'c)$

If both roots are same then

$\frac {a} {a'}=\frac {b} {b'}=\frac {c} {c'}$

Example :

Find p and q such that px2 +5x+2=0 and 3x2 +10x+q=0. have both roots in common?

Solution :

a1=p,b1=5,c1=2

a2 = 3, b2 = 10, c2 = q

We know that

$\frac {a} {a'}=\frac {b} {b'}=\frac {c} {c'}\, \, \, i.e\, \frac {p} {3}=\frac {5} {10}=\frac {2} {q}\, \, \, \, \, i.e\, p=3/2\, ;\, q\, =4$

Post By : Rahul Kumar 08 Apr, 2020 2672 views Maths