Swapping Symbols With Minimal Difference
A student notices that the roots of the equation are each one less than the roots of the equation .
Find the value of .
The answer is -4.
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Relevant wiki: Vieta's Formula - Quadratics - Basic
Let p and q be the roots of the 1st eqn. Naturally the roots of the 2nd eqn are p + 1 and q + 1 .
Now, p q = a and p + q = − b .
Also ( p + 1 ) ( q + 1 ) = b and p + 1 + q + 1 = − a which means p + q + 2 = − a
But we know p + q = − b so − b + 2 = − a Let this be as such, we'll come back to it later.
Now, ( p + 1 ) ( q + 1 ) = b ⟹ p q + p + q + 1 = b
Substituting values for p q and p + q again here, we get a − b + 1 = b ⟹ a = 2 b − 1 ⟹ − a = 1 − 2 b
Plugging it the 'marked for review' equation, we get, − b + 2 = 1 − 2 b ⟹ b = − 1
Since a = 2 b − 1 we get a = − 3
Therefore, a + b = − 3 − 1 = − 4