Philip P. answered • 06/14/14
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The graph of a quadratic equation is always a parabola. The x-intercepts are the points where the parabola crosses the x-axis. Will a parabola always cross the x-axis? No! Picture a parabola that opens upward. If its vertex lies above the x-axis, the parabola will never cross the x-axis. Hence it will have no x-intercepts. Similarly, if the parabola opens downwards (inverted) and its vertex lies below the x-axis, it will never cross the x-axis.
So to figure out whether a quadratic equation has 2, 1, or no solutions, you need to know two things: 1) the y-coordinate of the vertex and whether the parabola opens up or down (inverted).
For a general quadratic of the form ax2 + bx + c, the x-coordinate of the vertex is located at -b/2a. Plug that into the quadratic equation to get the y-coordinate of the vertex (yvertex)
Whether the parabola opens up or down depends on the sign of a, the coefficient of the x2 term. If a > 0, the parabola opens upward with the vertex at the bottom. If a < 0, the parabola opens downward (inverted) so the vertex is at the top.
Here are the rules for figuring out how many solutions a quadratic equation has:
- If the vertex lies above the x-axis (yvertex > 0) and the parabola opens upwards (a > 0), then the parabola has no x-intercepts.
- If the vertex lies below the x-axis (yvertex < 0) and the parabola opens downward (a < 0), then the parabola has no x-intercepts.
- If the vertex lies on the x-axis (yvertex=0), then the parabola has exactly one x-intercept and it's the vertex.
- For all other cases, the parabola will have two x-intercepts. You find them by factoring the quadratic equation or by using the quadratic formula.
Here's a link to my study guide on graphing quadratic equations.
http://www.wyzant.com/resources/files/263613/how_to_graph_a_quadratic_equation
