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If the parabola y = x^2 + 2x + 7 and the line y = 3x + b intersect at only one point, what is the value of b?

 Oct 12, 2020
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y = x^2 + 2x + 7  is a bowl shaped parabola.....the only ways a line can intersect it in ONE point is if the line is VERTICAL (this one has a slope of 3) ...or if the line is sloped , it must be tangent to the parabola   ( a third way would be for the line to be horizontal (slope = 0) and tangent at vertex)

When the line is tangent to the parabola, the parabola will have the same slope as the line ( 3)

 

find where the slope = 3

derivative of the parabola equation

   2x + 2       is the slope at each point x .....   eqaute to 3

2x+2 = 3

 x = 1/2    then   y = 8.25

 

so the line has the point   (1/2 , 8.25 )

plug these into the line equation

8.25 = 3(1/2) + b           b = 6.75

 Oct 12, 2020
edited by ElectricPavlov  Oct 12, 2020

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