 # 11.1 Solving Quadratic Equations by the Square Root Property

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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:

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

Solving ax2 + bx + c = 0 (continued): Write as a square: Use square root property: 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

Example:

Complex Numbers and the Quadratic Formula Solve x2 – 2x + 2 = 0

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

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

Example: Let u = x2

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

11.5 Formulas and Applications
Example:

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

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)

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)

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)

Vertex Formula: The graph of f(x) = ax2 + bx + c has vertex

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)