Question 415558
The equation has to be in the form {{{y = ax^2 + bx + c}}}
There are 2 roots. You can have:
(1) 2 real roots
(2) 1 real root
(3) 2 imaginary roots
(4) no roots
All this information can be gotten out of the quadratic formula:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
notice the +- before the square root sign
You could express the formula as
{{{x[1] = (-b + sqrt( b^2-4*a*c ))/(2*a) }}} 
and
{{{x[2] = (-b - sqrt( b^2-4*a*c ))/(2*a) }}} 
These are the x-coordinates of the 2 roots of the equation
example:
{{{2x^2 + 5x + 3 = 0}}}
The factors will look like {{{ (x - x[1])*(x - x[2]) }}}
{{{a = 2}}}
{{{b = 5}}}
{{{c = 3}}}
{{{x[1] = (-b + sqrt( b^2-4*a*c ))/(2*a) }}} 
{{{x[1] = (-5 + sqrt( 5^2-4*2*3 ))/(2*2) }}} 
{{{x[1] = (-5 + sqrt( 25 - 24 ))/4 }}}
{{{x[1] = (-5 + 1 )/4 }}} 
{{{x[1] = -1}}}
and the other root is
{{{x[2] = (-5 - sqrt( 25 - 4*2*3 ))/(2*2) }}} 
{{{x[2] = (-5 - sqrt( 25 - 24 ))/4 }}} 
{{{x[2] = (-5 - 1)/4 }}} 
{{{x[2] = -3/2}}}
Now I can rewrite what I found as:
{{{x - x[1] = 0}}}
{{{x - x[2] = 0}}}
and
{{{x + 1 = 0}}}
{{{x + (3/2) = 0}}}
So, the factors of the equation are:
{{{ (x + 1)*(x + 3/2) = 0}}}
{{{x^2 + x + (3/2)*x + 3/2 = 0}}}
{{{x^2 + (5/2)*x + 3/2 = 0}}} 
Multiply both sides by {{{2}}}
{{{2x^2 + 5x + 3 = 0}}} 
which is the equation, so the roots are -1 and -3/2
Here is the plot:
{{{ graph( 400, 400, -4, 4, -2, 8, 2x^2 + 5x + 3) }}}