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11.1 Solving Quadratic Equations by the Square Root Property

11.1 Solving Quadratic Equations by the Square Root Property. Square Root Property of Equations: If k is a positive number and if a 2 = k, then and the solution set is:. 11.1 Solving Quadratic Equations by the Square Root Property. Example:.

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11.1 Solving Quadratic Equations by the Square Root Property

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  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, thenand 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 realIf 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 FormulaSolve x2 – 2x + 2 = 0

  12. 11.4 Equations Quadratic in Form

  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 = 0Solve 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) 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 + his 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 ParabolasHorizontal Parabola with Vertex (h, k) Vertex = (h, k)

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