Question 937297: If the difference of the roots of 2xx-6x+c=0 is 5, find the roots and the value of c.
Answer by TimothyLamb(4379)   (Show Source):
You can put this solution on YOUR website! 2xx - 6x + c = 0
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complete the square:
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(2xx - 6x + c = 0)/2
xx - 3x + c/2 = 0
xx - 3x + (-3/2)^2 - (-3/2)^2 = -c/2
xx - 3x + (-3/2)^2 = -c/2 + (-3/2)^2
(x - 3/2)^2 = -c/2 + 9/4
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solve for x:
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sqrt( (x - 3/2)^2 ) = sqrt( -c/2 + 9/4 )
x - 3/2 = sqrt( -c/2 + 9/4 )
x = sqrt( -c/2 + 9/4 ) + 3/2
x = sqrt( -2c/4 + 9/4 ) + 3/2
x = sqrt( (9 - 2c)/4 ) + 3/2
x = sqrt( 9 - 2c )/2 + 3/2
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roots are:
x = 3/2 + sqrt( 9 - 2c )/2
x = 3/2 - sqrt( 9 - 2c )/2
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difference of roots is 5:
3/2 + sqrt( 9 - 2c )/2 - ( 3/2 - sqrt( 9 - 2c )/2 ) = 5
3/2 + sqrt( 9 - 2c )/2 - 3/2 + sqrt( 9 - 2c )/2 = 5
sqrt( 9 - 2c )/2 + sqrt( 9 - 2c )/2 = 5
sqrt( 9 - 2c ) = 5
( sqrt( 9 - 2c ) )^2 = 5*5
9 - 2c = 25
2c = 9 - 25
c = -16/2
c = -8
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roots:
x = 3/2 + sqrt( 9 - 2*(-8) )/2
x = 3/2 - sqrt( 9 - 2*(-8) )/2
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x = 4
x = -1
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