SOLUTION: how to solve for slope without using derivatives? please help :( y=-8/√x at points (4, -4)

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Question 1107802: how to solve for slope without using derivatives? please help :(
y=-8/√x at points (4, -4)
Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.
            The way is THIS: 

Consider the system of 2 equations 

y = -8%2Asqrt%28x%29,              (1)

y - (-4) = m(x-4).       (2)



It is the same, as the system

y = -8%2Asqrt%28x%29,              (1)

y + 4 = m(x-4).          (2)



Find  the slope "m" under the condition that the system has a UNIQUE solution (x,y) = (4,-4).


For it, substitute expression for y,  y = m*(x-4)-4  into equation (1). You will get

m*(x-4) + 4 = -8%2Asqrt%28x%29.       (*)


Introduce new variable  u = sqrt%28x%29.  Then the equation (*) takes the form

m%2A%28u%5E2-4%29+%2B+4 = -8%2Au,

m%2Au%5E2+-+4%2Am+%2B+4 = -8%2Au,

m%2Au%5E2+%2B+4%2Au+%2B+4 = 0.             (**)


The condition that the equation (**) has a unique solution in "u"  means that the discriminant of the equation (**) is equal to zero.


Write the discriminant of the quadratic equation (**) and equate it to zero:

d = b%5E2+-+4ac = 4%5E2+-+4%2Am%2A4 = 16 - 16m  ====>

the equation d = 0  becomes  16 - 16m = 0,  which implies  16m = 16  ====>  m = 1.


Answer.  The slope under the question is  m= 1,
      
         found without using derivatives.