Question 246206
Is this question part of a calculus course? The answer
is easiest with calculus.
{{{f(x) = 3x^2 + 4x - 2}}}
{{{fprime(x) = 6x + 4}}}
{{{g(x) =mx - 5}}}
{{{gprime(x) = m}}}
I want to know all solutions {{{x[1],y[1]}}}
where
 {{{fprime(x[1]) = gprime(x[1])}}}
{{{6x + 4 = m}}}
Also, {{{x[1],y[1]}}} is a solution for
{{{f(x) = g(x)}}}
{{{3x^2 + 4x - 2 = mx - 5}}}
{{{3x^2 + 4x - 2 = (6x + 4)*x - 5}}}
{{{3x^2 + 4x - 2 = 6x^2 + 4x - 5}}}
{{{3x^2 = 3}}}
{{{x^2 = 1}}}
{{{x = 1}}}
{{{x = -1}}}
and
{{{fprime(1) = 6*1 + 4}}}
{{{fprime(1) = 10}}}
{{{gprime(1) = m}}}
since 
{{{fprime(1) = gprime(1)}}}
{{{m = 10}}} 
{{{fprime(-1) = 6*(-1) + 4}}}
{{{fprime(-1) = -2}}}
{{{gprime(-1) = m}}}
{{{m = -2}}}
I'll do a plot of the quadratic and 2 lines:
{{{ graph( 600, 600, -4, 4, -7, 8, 3x^2 + 4x - 2, 10x - 5,-2x - 5 ) }}}