First slide

Theory of equations

Question

If $α, β$ be the roots of the equation $(x - a) (x - b) + c = 0 (c \neq 0),$ then the roots of the equation $(x - c - α) (x - c - β) = c,$ are

Moderate

A

$a$ and $b + c$
B

$a + c$ and $b$
C

$a + c$ and $b + c$
D

$a - c$ and $b - c$

Solution

We have,
$\begin{array}{l} (x - a) (x - b) + c = 0 \\ \Rightarrow x^{2} - x (a + b) + a b + c = 0 \end{array}$
Since $α, β$ are roots of this equation.
$∴ α + β = a + b$ and $α β = a b + c .$
Now,
$\begin{array}{l} (x - c - α) (x - c - β) = c \\ \Rightarrow (x - c)^{2} - (x - c) (α + β) + α β - c = 0 \\ \Rightarrow (x - c)^{2} - (x - c) (a + b) + a b + c - c = 0 \\ \Rightarrow (x - c)^{2} - (x - c) (a + b) + a b = 0 \end{array}$
$\Rightarrow ((x - c) - a) ((x - c) - b) = 0 \Rightarrow x = c + a$ and $x = c + b$
Thus, the given equation has $c + a$ and $c + b$ as its roots.

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