Question 177348
the roots of the equation are the value of x when y = 0.
this equates to the point where the graph crosses the x-axis.
that's why when you want to solve for the roots of a quadratic equation, you set the equation equal to 0.
if the graph doesn't cross the x-axis, then it doesn't have real roots.
in that case, it will have imaginary roots.
the values you see on the graph conform to the values you calculate.
the general form of a quadratic equation is:
{{{a*x^2 + b*x + c = y}}}
to find the roots you set y equal to 0.
you get:
{{{a*x^2 + b*x + c = 0}}}
to find the roots, you use the formula:
{{{x = (-b +- sqrt(b^2 - 4*a*c))/(2*a)}}}
if {{{sqrt(b^2 - 4*a*c)}}} is positive, then you have real roots.
if it is negative, then you have imaginary roots.
you will only see the graph cross the x-axis if it has real roots.
here's a graph that has real roots.
{{{x^2 + 2*x - 35}}}
the graph looks like this:
{{{graph(800,300,-10,+10,-40,20,x^2+2*x-35)}}}
the roots are x = 5, and x = -7
here's a graph that has imaginary roots.
{{{x^2 + 2*x + 2}}}
the graph looks like this:
{{{graph(800,300,-10,+10,-20,20,x^2+2*x+2)}}}
the graph with imaginary roots did not cross the x-axis.