Question:
Let $ \alpha, \beta $ be the roots of the equation,$(x - a) (x - b) = c, c \ne 0 $.Then the roots of the equation $ (x - \alpha) \, (x - \beta) + c = 0 $ are
Let $ \alpha, \beta $ be the roots of the equation,$(x - a) (x - b) = c, c \ne 0 $.Then the roots of the equation $ (x - \alpha) \, (x - \beta) + c = 0 $ are
Updated On: Jun 14, 2022
- a, c
- b, c
- a, b
- a + c, b + c
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The Correct Option is C
Solution and Explanation
Given, $ \alpha, \beta $ are the roots of (x - a) (x - b) - c = 0
$\Rightarrow (x - a) (x - b) - c = (x - \alpha) \, (x - \beta)$
$\Rightarrow (x - a) \, (x - b) = (x - \alpha) \, (x - \beta) + c $
$\Rightarrow $ a, b are the roots of equation $ (x - \alpha) \, (x - \beta) + c $
$\Rightarrow (x - a) (x - b) - c = (x - \alpha) \, (x - \beta)$
$\Rightarrow (x - a) \, (x - b) = (x - \alpha) \, (x - \beta) + c $
$\Rightarrow $ a, b are the roots of equation $ (x - \alpha) \, (x - \beta) + c $
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
