Question

if α , β are the roots of equation x²+5x+5=0 ,then equation whose roots are α+1 , and β+1 is
(a) x²+5x-5=0
(b) x²+3x+5=0
(c) x²+3x+1=0
(d) none of these

Answer 1

$\huge{\underline{\underline{\red{\mathfrak{Answer :}}}}}$

Given :

Equation is x² + 5x + 5

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To Find :

An equation whose zeroes are α + 1 , β + 1

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Solution :

$\Large \star {\underline{\boxed{\sf{Sum \: of \: zeros \: = \: \frac{-b}{a}}}}}$

Put Values

⇒ α + β = -b/a

⇒ α + β = -5/1

⇒ α + β = -(+5)

★ α + β = - 5.........(1)

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$\Large \star {\underline{\boxed{\sf{Product \: of \: zeros \: = \: \frac{c}{a}}}}}$

Put Values

⇒ α.β = c/a

⇒ α.β = 5/1

★ α.β = 5........(2)

$\rule{200}{2}$

Zeros are α+1 , β + 1

Now, again use formula for sum of zeros

⇒ α + 1 + β + 1 = -b/a

⇒ α + β + 2 = -b/a

Put value of (1)

⇒ -5 + 2 = -b/a

⇒ -3 = -b/a

★ b/a = 3 (Sum of zeros = 3)

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Now take formula for product of zeroes

⇒ (α + 1)(β + 1) = c/a

⇒ (αβ + β + α + 1) = c/a

Put value of (1) and (2)

⇒ - 5 + 1 = c/a

★ c/a = 1 (Product of zeroes = 1)

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We have formula for equation,

${\underline{\boxed{\sf{Equation \: = \: x^2 \: - \: (Sum \: of \: zeros)x \: + \: Product \: of \: zeroes}}}}$

Put Values

⇒x² - 3x + 1

★ Equation with zeros as α + 1 and β + 1 is x² - 3x + 1

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