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Question 166875: Please help me. I'm stumped on 2 problems in Algebra 2.
Here they are:
1. Find the dimensions of a rectangle "a" with the greatest area whose perimeter is 30 feet.
2. Given x^3-4x^2+2x+1=0
(a) How many possible positive roots are there?
(b) How many possible negative roots are there?
(c) What are the possible rational roots?
(d) Using synthetic substitution, which of the possible rational roots is actually a root of the equation?
(e) Find the irrational roots of the equation. (Hint: Use the quadratic formula to solve the depressed equaiton.)
Thanks a lot!
Andrew
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 1. Find the dimensions of a rectangle "a" with the greatest area whose perimeter is 30 feet.
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P = 2l +2w
30 = 2l + 2w
15 = l + w
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Area = l(w)
A = w(15-w)
A = 15w - w^2
This is a quadratic with a=-1,b=15,c= -A
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Greatest area occurs when w = -b/2a = -15/(2(-1)) = 7.5 feet (this is the width)
Since L+w = 15, length = 7.5 ft
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2. Given x^3-4x^2+2x+1=0
# of sign changes in f(x) = 2
# of sign changes in f(-x) = 1
(a) How many possible positive roots are there?: 0 or 2
(b) How many possible negative roots are there?: 1
(c) What are the possible rational roots? 1 and -1
(d) Using synthetic substitution, which of the possible rational roots is actually a root of the equation? +1
(e) Find the irrational roots of the equation. (Hint: Use the quadratic formula to solve the depressed equation.)
1)....1....-4....2....1
-------1-----3....-1..|..0
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Quotient: x^2-3x-1
Roots: x = [3 +- sqrt(9 - 4*1*-1)]/2
x = [3 +- sqrt(13)]/2
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Cheers,
Stan H.
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