A value of b for which the equations x2+bx-1=0,x2+x+b=0 have one root in common is

Question

A value of b for which the equations x2+bx-1=0,x2+x+b=0 have one root in common is (A) -√2 (B) -i√3 (C) i√5 (D) √2

Tardigrade Admin · Accepted Answer

If a1 x2+b1 x+c1=0 and a2 x2+b2 x+c2=0 have a common real root, then ⇒(a1 c2-a2 c1)2=(b1 c2-b2 c1)(a1 b2-a2 b1) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/7a2881424880e00b863ed060172cbe6f-.png" /> have a common root ⇒(1+b2)=(b2+1)(1-b) ⇒ b2+2 b+1=b2-b3+1-b ⇒ b3+3 b=0 ∴ b(b2+3)=0 ⇒ b=0, ± √3 i

Q. A value of $b$ for which the equations $x^{2} + b x - 1 = 0, x^{2} + x + b = 0$ have one root in common is

$- 2$

$- i 3$

$i 5$

$2$

Q. A value of b for which the equations x2+bx−1=0,x2+x+b=0 have one root in common is

−2​

−i3​

i5​

2​

If a1​x2+b1​x+c1​=0 and a2​x2+b2​x+c2​=0 have a common real root, then ⇒(a1​c2​−a2​c1​)2=(b1​c2​−b2​c1​)(a1​b2​−a2​b1​) have a common root ⇒(1+b2)=(b2+1)(1−b) ⇒b2+2b+1=b2−b3+1−b ⇒b3+3b=0 ∴b(b2+3)=0⇒b=0,±3​i

Q. A value of $b$ for which the equations $x^{2} + b x - 1 = 0, x^{2} + x + b = 0$ have one root in common is

$- 2$

$- i 3$

$i 5$

$2$