Question

If α and β are the roots of the equation x² + 5x + 5 =0 . Then Find the Equation whose roots are (α + 1) and (β + 1).

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Answer 1

$\LARGE\underbrace{\purple{\sf{Required \; Solution:}}}$

The quadratic equation whose roots are α + 1, β + 1 is x² + 3x + 1 = 0

• Given equation is

 ⠀⠀⠀⠀$\sf{x^2 + 5x + 5 = 0}$ ——— (1)

• Comparing with ax² + bx + c = 0, we get

 ⠀⠀⠀⠀$\sf{a = 1, \; b = 5, \; c = 5}$

• Roots of (1) are α, β

 ⠀⠀⠀⠀$\sf{sum \; of \; roots = \alpha + \beta = \dfrac{-b}{a} = \dfrac{-5}{1} = -5}$ ——— (2)

 ⠀⠀⠀⠀$\sf{product \; of \; roots = \alpha\beta = \dfrac{c}{a}=\dfrac{5}{1} = 5}$——— (3)

• Quadratic equation with roots α + 1, β + 1 is given by

 ⠀⠀⠀⠀x² - (sum of roots)x + (product of roots) = 0

 ⠀⠀⠀⠀x² - (α + 1 + β + 1)x + (α+1)(β+1) = 0

 ⠀⠀⠀⠀x² - (α + β + 2)x + αβ + α + β + 1 = 0

 ⠀⠀⠀⠀x² - (-5+2)x + 5 + (-5) + 1 = 0       [from (2) and (3)]

 ⠀⠀⠀⠀x² - (-3)x + 5 - 5 + 1 = 0

 ⠀⠀⠀⠀x² + 3x + 1 = 0

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