11.1 Solving Quadratic Equations by the Square Root Property Square Root Property of Equations: If k is a positive number and if a2 = k, then and the solution set is:

11.1 Solving Quadratic Equations by the Square Root Property Example:

11.2 Solving Quadratic Equations by Completing the Square Example of completing the square:

11.2 Solving Quadratic Equations by Completing the Square Completing the Square (ax2 + bx + c = 0): Divide by a on both sides (lead coefficient = 1) Put variables on one side, constants on the other. Complete the square (take ½ of x coefficient and square it – add this number to both sides) Solve by applying the square root property

11.2 Solving Quadratic Equations by Completing the Square Review: x4 + y4 – can be factored by completing the square

11.2 Solving Quadratic Equations by Completing the Square Example: Complete the square: Factor the difference of two squares:

11.3 Solving Quadratic Equations by the Quadratic Formula Solving ax2 + bx + c = 0: Dividing by a: Subtract c/a: Completing the square by adding b2/4a2:

11.3 Solving Quadratic Equations by the Quadratic Formula Solving ax2 + bx + c = 0 (continued): Write as a square: Use square root property: Quadratic formula:

11.3 Solving Quadratic Equations by the Quadratic Formula Quadratic Formula: is called the discriminant. If the discriminant is positive, the solutions are real If the discriminant is negative, the solutions are imaginary

11.3 Solving Quadratic Equations by the Quadratic Formula Example:

11.3 Solving Quadratic Equations by the Quadratic Formula Complex Numbers and the Quadratic Formula Solve x2 – 2x + 2 = 0

11.4 Equations Quadratic in Form Method Advantages Disadvantages Factoring Fastest method Not always factorable Square root property Not always this form Completing the square Can always be used Requires a lot of steps Quadratic Formula Slower than factoring

11.4 Equations Quadratic in Form Sometimes a radical equation leads to a quadratic equation after squaring both sides An equation is said to be in “quadratic form” if it can be written as a[f(x)]2 + b[f(x)] + c = 0 Solve it by letting u = f(x); solve for u; then use your answers for u to solve for x

11.4 Equations Quadratic in Form Example: Let u = x2

11.5 Formulas and Applications Example (solving for a variable involving a square root)

11.5 Formulas and Applications Example:

11.6 Graphs of Quadratic Functions A quadratic function is a function that can be written in the form: f(x) = ax2 + bx + c The graph of a quadratic function is a parabola. The vertex is the lowest point (or highest point if the parabola is inverted

11.6 Graphs of Quadratic Functions Vertical Shifts: The parabola is shifted upward by k units or downward if k < 0. The vertex is (0, k) Horizontal shifts: The parabola is shifted h units to the right if h > 0 or to the left if h < 0. The vertex is at (h, 0)

11.6 Graphs of Quadratic Functions Horizontal and Vertical shifts: The parabola is shifted upward by k units or downward if k < 0. The parabola is shifted h units to the right if h > 0 or to the left if h < 0 The vertex is (h, k)

11.6 Graphs of Quadratic Functions Graphing: The vertex is (h, k). If a > 0, the parabola opens upward. If a < 0, the parabola opens downward (flipped). The graph is wider (flattened) if The graph is narrower (stretched) if

11.6 Graphs of Quadratic Functions Inverted Parabola with Vertex (h, k)

11.7 More About Parabolas; Applications Vertex Formula: The graph of f(x) = ax2 + bx + c has vertex

11.7 More About Parabolas; Applications Graphing a Quadratic Function: Find the y-intercept (evaluate f(0)) Find the x-intercepts (by solving f(x) = 0) Find the vertex (by using the formula or by completing the square) Complete the graph (plot additional points as needed)

11.7 More About Parabolas; Applications Graph of a horizontal (sideways) parabola: The graph of x = ay2 + by + c or x = a(y - k)2 + h is a parabola with vertex at (h, k) and the horizontal line y = k as axis. The graph opens to the right if a > 0 or to the left if a < 0.

11.7 More About Parabolas Horizontal Parabola with Vertex (h, k)