The value of c (≠ 1) for which the equation x 2+ x + c =0 and x 2+ cx +1=0 have one root in common is

Question

The value of c (≠ 1) for which the equation x 2+ x + c =0 and x 2+ cx +1=0 have one root in common is (A) 2 (B) -1 (C) -2 (D) 0

Tardigrade Admin · Accepted Answer

Let α be their common root. So, α2+α+ c =0 ....(1) and α2+ c α+1=0 ...(2) Subtracting (1-c) α+(c-1)=0 ⇒(1-c)(α-1)=0 But c ≠ 1 ⇒ α=1 ∴ Putting α=1 in equation (1), we get 1+1+ c =0 Hence, c =-2.

Q. The value of $c (\neq = 1)$ for which the equation $x^{2} + x + c = 0$ and $x^{2} + c x + 1 = 0$ have one root in common is

2

-1

-2

0

Let $α$ be their common root.
So, $α^{2} + α + c = 0$ ....(1)
and $α^{2} + c α + 1 = 0$ ...(2)
Subtracting _________
$(1 - c) α + (c - 1) = 0 \Rightarrow (1 - c) (α - 1) = 0$
But $c \neq = 1 \Rightarrow α = 1$
$∴$ Putting $α = 1$ in equation (1), we get $1 + 1 + c = 0$
Hence, $c = - 2$ .

Q. The value of c(=1) for which the equation x2+x+c=0 and x2+cx+1=0 have one root in common is

2

-1

-2

0

Let α be their common root. So, α2+α+c=0 ....(1) and α2+cα+1=0 ...(2) Subtracting _________ (1−c)α+(c−1)=0⇒(1−c)(α−1)=0 But c=1⇒α=1 ∴ Putting α=1 in equation (1), we get 1+1+c=0 Hence, c=−2.

Q. The value of $c (\neq = 1)$ for which the equation $x^{2} + x + c = 0$ and $x^{2} + c x + 1 = 0$ have one root in common is

Let $α$ be their common root.
So, $α^{2} + α + c = 0$ ....(1)
and $α^{2} + c α + 1 = 0$ ...(2)
Subtracting _________
$(1 - c) α + (c - 1) = 0 \Rightarrow (1 - c) (α - 1) = 0$
But $c \neq = 1 \Rightarrow α = 1$
$∴$ Putting $α = 1$ in equation (1), we get $1 + 1 + c = 0$
Hence, $c = - 2$ .