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If you were to graph a quadratic function, you would see that it results in a parabola that either opens upwards (positive x^2 coefficient) or downwards (negative x^2 coefficient). From the graph you can readily see that the parabola crosses the y-axis only once. But, depending on the sign of the x^2 coefficient and/or the value of the constant term, the parabola may or may not cross the x-axis.
If the parabola does cross the x-axis, it will cross in two places and the function will have two real roots.
If the parabola does not cross the x-axis, then the function will have two complex roots.
A couple of examples should illustrate these ideas:
1) Quadratic functions with two real roots: 1) y=x^2-2 and 2) y = -x^2+2
1) Quadratic functions with two complex roots: 1) y=x^2+2 and 2) y=-x^2-2
Finding the roots without graphing the function can be done in several different ways. The best method to use depends on how the function is written.
Some of the methods are:
2.Completing the square.
3.Using the quadratic formula,
Method 3. will always work.