1. 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:

2. 11.1 Solving Quadratic Equations by the Square Root Property
Example:

3. 11.2 Solving Quadratic Equations by Completing the Square
Example of completing the square:

4. 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

5. 11.2 Solving Quadratic Equations by Completing the Square
Review:
x4 + y4 – can be factored by completing the square

6. 11.2 Solving Quadratic Equations by Completing the Square
Example: Complete the square: Factor the difference of two squares:

7. 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:

8. 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:

9. 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

10. 11.3 Solving Quadratic Equations by the Quadratic Formula
Example:

11. 11.3 Solving Quadratic Equations by the Quadratic Formula
Complex Numbers and the Quadratic Formula Solve x2 – 2x + 2 = 0

12. 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

13. 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

14. 11.4 Equations Quadratic in Form
Example: Let u = x2

15. 11.5 Formulas and Applications
Example (solving for a variable involving a square root)

16. 11.5 Formulas and Applications
Example:

17. 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

18. 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)

19. 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)

20. 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

21. 11.6 Graphs of Quadratic Functions Inverted Parabola with Vertex (h, k)

22. 11.7 More About Parabolas; Applications
Vertex Formula: The graph of f(x) = ax2 + bx + c has vertex

23. 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)

24. 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.

25. 11.7 More About Parabolas Horizontal Parabola with Vertex (h, k)