I have two functions and want to find out in which points they intersect f ( x ) = − 3 x 3 − 3 x 2 + A x g ( x ) = x 3 + x 2 − 6 xI know that parameter should be A = 18, but it was just my guess. What I want to as you for is the process how to find it.I know that I must start by f ( x ) = g ( x ) and then f ( x ) − g ( x ) = 0 and so on, but I am stuck with quadratic equation with parameter.

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I have two functions and want to find out in which points they intersect  f  (  x  )  =  −  3      x    3    −  3      x    2    +  A  x  g  (  x  )  =      x    3    +      x    2    −  6  xI know that parameter should be   A  =  18, but it was just my guess. What I want to as you for is the process how to find it.I know that I must start by   f  (  x  )  =  g  (  x  ) and then   f  (  x  )  −  g  (  x  )  =  0 and so on, but I am stuck with quadratic equation with parameter.

Messiah Trevino · Accepted Answer

Step 1  y  =  f  (  x  ) and   y  =  g  (  x  ) intersect where   f  (  x  )  =  g  (  x  ). Here that is   −  3      x    3    −  3      x    2    +  A  x  =      x    3    +      x    2    −  6  x. The first thing I see is that   x  (  −  3      x    2    −  3  x  +  A  )  =  x  (      x    2    +  x  −  6  ) so   x  =  0 is one solution. If   x  ≠  0 then we can divide both sides by x to get   −  3      x    2    −  3  x  +  A  =      x    2    +  x  −  6. We can write that as   4      x    2    +  4  x  −  (  A  +  6  )  =  0. That is a quadratic equation which has roots   x  =            −      4      ±              16        −        4        (        4        )        (        A        +        6        )                    2      (      4      )        =            −      4      ±      4              1        −        A        −        6              8  Step 2Now the question of the number of real roots depends upon the value of   −  A  −  5. If   A  =  −  5 then   −  A  −  5  =  0 so the quadratic equation has one real solution and the graphs cross twice. If   A  &amp;lt;  −  5 so   −  A  −  5  &amp;gt;  0 the quadratic equation has two real solutions and the graphs cross three times. If   A  &amp;gt;  −  5 so   −  A  −  5  &amp;lt;  0 so the quadratic equation has no real solutions and the graphs cross only once.

I have two functions and want to find out in which points they intersect. f(x)=-3x^3-3x^2+Ax. g(x)=x^3+x^2-6x.

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