How to Solve and Graph a Quadratic Equation

Definition of a Quadratic Equation

• A quadratic equation is typically written in the form: [ ax^2 + bx + c = 0 ] where:
  • ( a ), ( b ), and ( c ) are constants
  • ( a ) cannot be 0

Methods to Solve Quadratic Equations

1. Factoring

   • Express the quadratic in the form ( (px + q)(rx + s) = 0 )
   • Solve for ( x ) by setting each factor equal to zero.

2. Completing the Square

   • Adjust the equation to be in the form ( (x - p)^2 = q )
   • Solve for ( x ) by taking the square root of both sides.

3. Quadratic Formula

   • Use the formula: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
   • Discriminant ( (b^2 - 4ac) ) indicates the nature of roots:
     • Positive: Two real solutions
     • Zero: One real solution
     • Negative: No real solutions (complex roots)

Graphing a Quadratic Equation

• Graphs of quadratic equations are called parabolas.
• Parabolas can open upwards or downwards.

Key Features of a Parabola

1. Vertex:

   • The highest or lowest point of the parabola.
   • Found using the formula: ( x = -\frac{b}{2a} )
   • Substitute back into the equation to find the y-coordinate.

2. Axis of Symmetry:

   • A vertical line passing through the vertex ( x = -\frac{b}{2a} )

3. Y-intercept:

   • Point where the graph crosses the y-axis ((x = 0))
   • Found by calculating ( c ) in the equation ( ax^2 + bx + c )

4. X-intercepts (Roots):

   • Points where the graph crosses the x-axis
   • Found by solving ( ax^2 + bx + c = 0 )

Graphing Steps

1. Find the Vertex

   • Calculate ( x = -\frac{b}{2a} ) and substitute into the equation to get ( y ).

2. Find the Axis of Symmetry

   • Use the vertex to identify the line of symmetry.

3. Identify the y-intercept

   • Calculate ( c ).

4. Determine the x-intercepts

   • Solve the quadratic equation.

5. Plot Points

   • Plot the vertex, axis of symmetry, and intercepts.
   • Draw a smooth curve through the points to form the parabola.

Additional Resources

• Viewing videos or tutorials on quadratic graphing methods can enhance understanding (e.g., Khan Academy, YouTube tutorials).

Remember, practice is key in mastering the solving and graphing of quadratic equations. Utilizing graphing calculators or software can also be beneficial for visual learning.