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# First Quarter - Chapter 2 - Quadratic Equation

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First Quarter - Chapter 2 - Quadratic Equation

First Quarter - Chapter 2 - Quadratic Equation

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• 2. *1 – Definition of Quadratic Equation2 – Solving Quadratic Equations by the Square Root Property3 – Solving Quadratic Equations by Completing the Square4 – Solving Quadratic Equations by the Quadratic Formula5 – Graphing Quadratic Equations in Two Variables6 – Interval Notation, Finding Domains and Ranges from Graphs and Graphing Piecewise-Defined Functions
• 3. *Quadratic Equations*An example of a Quadratic Equation:*The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).*It is also called an "Equation of Degree 2" (because of the "2" on the x)
• 4. The Standard Form of a Quadratic Equation looks like this: The letters a, b and c are coefficients (you know those values). They can have any value, except that a cant be 0. The letter "x" is the variable or unknown (you dont know it yet)
• 5. Solving Quadratic Equationsby the Square Root Property
• 6. *We previously have used factoring to solve quadraticequations.This chapter will introduce additional methods for solvingquadratic equations.Square Root Property If b is a real number and a2 = b, then a b
• 7. Example Solve x2 = 49 x 49 7 Solve 2x2 = 4 x2 = 2 x 2 Solve (y – 3)2 = 4 y 3 4 2 y=3 2 y = 1 or 5
• 8. Example Solve x2 + 4 = 0 x2 = 4 There is no real solution because the square root of 4 is not a real number.
• 9. Example Solve (x + 2)2 = 25 x 2 25 5 x= 2 5 x = 2 + 5 or x = 2 – 5 x = 3 or x = 7
• 10. Example Solve (3x – 17)2 = 28 3x – 17 = 28 2 7 3x 17 2 7 17 2 7 x 3
• 11. Solving QuadraticEquations by Completingthe Square
• 12. Completing the SquareIn all four of the previous examples, theconstant in the square on the right side, is halfthe coefficient of the x term on the left.Also, the constant on the left is the square ofthe constant on the right.So, to find the constant term of a perfect squaretrinomial, we need to take the square of half thecoefficient of the x term in the trinomial (as longas the coefficient of the x2 term is 1, as in ourprevious examples).
• 13. Example What constant term should be added to the following expressions to create a perfect square trinomial? x – 10x 2 add 52 = 25 x2 + 16x add 82 = 64 x2 – 7x 2 7 49 add 2 4
• 14. Example We now look at a method for solving quadratics that involves a technique called completing the square. It involves creating a trinomial that is a perfect square, setting the factored trinomial equal to a constant, then using the square root property from the previous section.
• 15. Solving a Quadratic Equation by Completing aSquare 1) If the coefficient of x2 is NOT 1, divide both sides of the equation by the coefficient. 2) Isolate all variable terms on one side of the equation. 3) Complete the square (half the coefficient of the x term squared, added to both sides of the equation). 4) Factor the resulting trinomial. 5) Use the square root property.
• 16. Solving EquationsExample Solve by completing the square. y2 + 6y = 8 y2 + 6y + 9 = 8 + 9 (y + 3)2 = 1 y+3= 1 = 1 y= 3 1 y = 4 or 2
• 17. Example Solve by completing the square. y2 + y – 7 = 0 y2 + y = 7 y2 + y + ¼ = 7 + ¼ 29 (y + ½)2 = 4 1 29 29 y 2 4 2 1 29 1 29 y 2 2 2
• 18. ExampleSolve by completing the square. 2x2 + 14x – 1 = 0 2x2 + 14x = 1 x2 + 7x = ½ 49 49 51 x2 + 7x + 4 =½+ 4 = 4 7 51 (x + )2 = 2 4 7 51 51 7 51 7 51 x x 2 4 2 2 2 2