Math 101 Intermediate Algebra    Solving Quadratic Equations by the Quadratic Formula Chapter 9, Section 2 The Quadratic Formula Given the problem Solve ax2 + bx + c = 0, the solutions are The quadratic formula is easily derived by completing the square to solve the equation ax2 + bx + c = 0. Using the Quadratic Formula Write the given quadratic equation in standard form ax2 + 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, b2 - 4ac The discriminant tells how many different values will be solutions. If b2 - 4ac > 0, the quadratic equation has 2 distinct real number solutions. If b2 - 4ac = 0, the quadratic equation has a single real number solution (repeated twice). If b2 - 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 = ax2 + 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 y-intercept: Find the value of y when x = 0. Find the x-intercepts: The x-intercepts are the points where y = 0. So, solve ax2 + bx + c = 0. The solutions are the x-coordinates of the x-intercepts. There may be 0, 1, or 2 x-intercepts. Find the vertex of the parabola The x-coordinate of the vertex is x = -b/2a. To find the y-coordinate of the vertex, plug the number x = -b/2a into y = ax2 + bx + c. Plot the y-intercept, the x-intercepts (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.