1. Solving Quadratic Equation by Graphing

2. Quadratic Solutions The number of real solutions is at most two.
No solutions
One solution
Two solutions

3. Solving Equations
When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts. These values are also referred to as solutions, zeros, or roots.

4. Identifying Solutions
Example f(x) = x2 - 4
Solutions are -2 and 2.

5. Identifying Solutions
Now you try this problem. f(x) = 2x - x2 Solutions are 0 and 2.

6. Graphing Quadratic Equations
The graph of a quadratic equation is a parabola. The roots or zeros are the x-intercepts. The vertex is the maximum or minimum point. All parabolas have an axis of symmetry.

7. Graphing Quadratic Equations
One method of graphing uses a table with arbitrary x-values. Graph y = x2 - 4x Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2 x y 1 -3 2 -4 3 4

8. Graphing Quadratic Equations
Try this problem y = x2 - 2x - 8. Roots Vertex Axis of Symmetry x y -2 -1 1 3 4

9. Graphing Quadratic Equations
The graphing calculator is also a helpful tool for graphing quadratic equations.

10. Using the ZERO command:
Solve: 
Since this equation is set equal to zero, the zeros (roots) will be the locations where the graph crosses the x-axis (if the roots are real numbers).
1.  Set 


11. Left Bound/Right Bound
2.  Use the ZERO command to find the zeros (roots) --  2nd TRACE (CALC), #2 zero
3.  Left bound?  Move the spider as close to the zero (root --- where the graph crosses the x-axis) as possible.  Hit the left arrow to move to the "left" of the zero (root).  Hit ENTER.  A "marker" ► will be set to the left of the zero (root).


12. Left Bound/Right Bound
4.  Right bound?  Move the spider as close to the zero (root --- where the graph crosses the x-axis) as possible.  Hit the right arrow to move to the "right" of the zero (root).  Hit ENTER.  A "marker" ◄ will be set to the right of the zero (root).

13. ZEROS/ROOTS/SOLUTIONS/ X-INTERCEPTS
5.  Guess?  Just hit ENTER.
6.  Repeat the entire process to find other zeros (roots --- which in this case the second root happens to be x = 7).
Answer:  one of the zeros (roots) is x = -2