The equations x2 - ax + b = 0 and x2 + bx - a = 0 have a common root, then

Question

The equations x2 - ax + b = 0 and x2 + bx - a = 0 have a common root, then (A) a = b (B) a + b = 1 (C) a + b = 0 or a-b=1 (D) a - b = 2.

Tardigrade Admin · Accepted Answer

Let α be a common root of the given equations. ∴ a2 - aα + b = 0 and a2 + bα - a= 0. ⇒ (a + b) α - (a + b) = 0 ⇒ (a + b) (α - 1) = 0 ⇒ a + b = 0 or α = 1 If α = 1, then 1 - a + b= 0 ⇒ a-b = 1 .

Q. The equations $x^{2} - a x + b = 0$ and $x^{2} + b x - a = 0$ have a common root, then

$a = b$

$a + b = 1$

$a + b = 0$ or $a - b = 1$

$a - b = 2.$

Let $α$ be a common root of the given equations.
$∴ a^{2} - a α + b = 0$ and $a^{2} + b α - a = 0$ .
$\Rightarrow (a + b) α - (a + b) = 0$
$\Rightarrow (a + b) (α - 1) = 0$
$\Rightarrow a + b = 0$ or $α = 1$
If $α = 1$ , then $1 - a + b = 0$
$\Rightarrow a - b = 1$ .

Q. The equations x2−ax+b=0 and x2+bx−a=0 have a common root, then

a=b

a+b=1

a+b=0 or a−b=1

a−b=2.