They are consecutive

Algebra Level 3

If roots of x 2 − a x + b = 0 x^2-ax+b=0 are two consecutive integers , then which of the following is true?

a 2 = 2 b + 1 a^2=2b+1 a 2 = 3 b − 1 a^2=3b-1 a 2 = 4 b + 1 a^2=4b+1 a 3 = a b + 2 a^3=ab+2

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A Former Brilliant Member by A Former Brilliant Member

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2 solutions

Rishabh Jain
Aug 9, 2016

Difference between roots( α , β \alpha,\beta ) of this equation is one,

∴ ( α − β ) 2 = 1 ⟹ ( α + β ) 2 − 4 α β = 1 ⟹ ( a ) 2 − 4 b = 1 ⟹ a 2 = 4 b + 1 \therefore (\alpha-\beta)^2=1\implies (\alpha+\beta)^2-4\alpha\beta=1\implies (a)^2-4b=1\implies a^2=4b+1

See Vieta's formula

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Akshat Sharda
Aug 9, 2016

Let one root be k k then other is k + 1 k+1 .

a = 2 k + 1 ⇒ k = a − 1 2 b = k ( k + 1 ) = k 2 + k = a 2 − 2 a + 1 4 + a − 1 2 b = a 2 − 1 4 ⇒ 4 b + 1 = a 2 a=2k+1\Rightarrow k=\frac{a-1}{2} \\ b=k(k+1)=k^2+k = \frac{a^2-2a+1}{4}+\frac{a-1}{2} \\ b = \frac{a^2-1}{4} \Rightarrow \boxed{4b+1=a^2}

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Nice solution. :)

A Former Brilliant Member - 4 years, 10 months ago

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