Question

If one root of the two quadratic equations x² + ax + b = 0 and x² + bx + a = 0 is common, then 

A) a + b = -1 

B) ab = -1 

C) ab = 1 

D) a + b = 1

Answer 1

Correct option is (A) a + b = -1

Let \(\alpha\) be common root of given quadratic equations.

\(\therefore\alpha^2+a\alpha+b=0\)        ______________(1)

& \(\alpha^2+b\alpha+a=0\)        ______________(2)

Subtract equation (2) from (1), we get

\((\alpha^2+a\alpha+b)-(\alpha^2+b\alpha+a)=0\)

\(\Rightarrow(a-b)\alpha+b-a=0\)

\(\Rightarrow(a-b)\alpha-(a-b)=0\)

\(\Rightarrow\alpha=\frac{a-b}{a-b}=1\)

Hence, x = 1 is a common root of given quadratic equations.

Put x = 1 in given equation, we get

\(1^2+a.1+b=0\)

\(\Rightarrow\) 1+a+b = 0

\(\Rightarrow\) a + b = -1

Answer 2

Correct option is A) a + b = -1