 The Quadratic Formula
 Given the problem
Solve ax^{2} + bx + c = 0,
the solutions are
 The quadratic formula is easily derived by completing the square
to solve the equation ax^{2} + bx + c = 0.
 Using the Quadratic Formula

 Write the given quadratic equation in standard form ax^{2} + bx + c = 0.
 Determine the numerical values for a, b, and c.
 Substitute the values for a, b, and c into the quadratic formula.
 Evaluate the formula to obtain the solutions.
 The Discriminant, b^{2}  4ac
 The discriminant tells how many different values will be solutions.

 If b^{2}  4ac > 0,
 the quadratic equation has 2 distinct real number solutions.
 If b^{2}  4ac = 0,
 the quadratic equation has a single real number solution (repeated twice).
 If b^{2}  4ac < 0,
 the quadratic equation has no real number solutions, it has 2 complex solutions.
 Sketching Parabolas

A parabola is the name of a graph with the equation
y = ax^{2} + bx + c.
 Steps to sketching the graph of a parabola
 Determine whether the parabola opens upward or downward:
 If a > 0, the parabola opens upward;
 If a < 0, the parabola opens downward.
 Find the yintercept:
 Find the value of y when x = 0.
 Find the xintercepts:
 The xintercepts are the points where y = 0.
 So, solve ax^{2} + bx + c = 0.
 The solutions are the xcoordinates of the xintercepts.
 There may be 0, 1, or 2 xintercepts.
 Find the vertex of the parabola
 The xcoordinate of the vertex is x = b/2a.
 To find the ycoordinate of the vertex, plug the number x = b/2a into y = ax^{2} + bx + c.
 Plot the yintercept, the xintercepts (if any), and the vertex.
 Sketch the graph by drawing a smooth curve, with the shape of parabola, through the points just plotted and opening in the direction found in step 1.
