Let P(x) be a quadratic polynomial such that the product of the roots of P is 20. Real numbers a and b satisfy a+b=22 and P(a)+P(b)=P(22). Find a2+b2.
P(x) = gx^2 + hx + c product of roots is c/g = 20, so c=20g
P(x) = gx^2 + hx + 20g
P(a)+P(b)=P(22)
ga^2 + ha + 20g + gb^2 + hb + 20g = g*22^2 + h*22 + 20g
g(a^2+b^2) + h(a+b) + 20g = 484g + 22h
a+b=22 so more cancellations
g(a^2+b^2) + 20g = 484g
a^2+b^2 = 464
Solving for a and b was not asked, but ...
a+b=22
a^2 + 2ab + b^2 = 484
a^2+b^2 = 464
2ab = 20
ab = 10
a+b=22
ab = 10
a + 10/a = 22
a^2 - 22a + 10 = 0
a = (22 ± √(484-40))/2
(a,b) = {11 + √111, 11 - √111}
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Posted by Larry
on 2025-05-07 12:39:03 |
Solution
