Problem 4 (The Case of the Two Quadratics)

The two roots of the quadratic equation

x² + p x + q = 0

are individually one more than the two roots of

x² - p x + q = 0

Find the roots of the equations and find the values of p and q.

Solution

It is easy to see that the roots of

x² + bx + c = 0

are

x₁ = -b/2 + sqrt (b² - 4c)
x₂ = -b/2 - sqrt (b² - 4c)
and the roots of

x² - bx + c = 0
are
x₁ = b/2 + sqrt (b² - 4c)
x₂ = b/2 - sqrt (b² - 4c)
Hence we have
-b/2 + sqrt (b² - 4c) + 1 = b/2 + sqrt (b² - 4c)
-b/2 - sqrt (b² - 4c) + 1 = b/2 - sqrt (b² - 4c)

We can easily see that both equations imply b = 1 with no restrictions on c. Thus b =1 and c is arbitrary.

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Last modified on Wednesday, March 17, 1999