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L6 Quadratic Equations and the Quadratic Formula; Applications Quadratic Equations A quadratic equation is an equation of the form ax 2 + bx + c = 0 where a, b, and c are real numbers and a ≠ 0 . Solving by Factoring Zero-Product Property: If ab = 0 , then Example: Solve by factoring. x 2 − 10 x = −9 4 x 2 − 12 x + 9 = 0 51 The Square Roots Method: The solution set to x 2 = p ( p ≥ 0) is the set of all square roots of the number p, that is x=± p Example. Solve by using the Square Roots Method. x 2 = 25 ( x + 2) 2 = 3 Solving Quadratic Equations by Completing the Square: ax 2 + bx + c = 0, a≠0 1. Make sure that a = 1; if not, divide each term by a. 2. Get the constant on the right-hand side of the equation. 3. Take 1/2 of the coefficient of x, square it, and add this number to both sides of the equation. 4. Factor the left-hand side into a perfect square. 5. Solve for x by using the Square Roots Method. 52 Example: Solve by completing the square. x2 − 8x + 3 = 0 Example: Factor by using completing the square. x2 + 4 x − 3 = 53 The Quadratic Formula The quadratic equation ax 2 + bx + c = 0 a≠0 has the solutions (roots) −b ± b 2 − 4ac x= 2a The quantity b 2 − 4ac , denoted D, is called the discriminant. The equation has: two distinct real roots if D > 0 one repeated real root if D = 0 no real solutions if D < 0 Note: If a < 0 , multiply both sides of the equation by a −1 to make use of the quadratic formula easier. The Half-coefficient Quadratic Formula: If in the quadratic equation ax 2 + bx + c = 0 , a≠0 b is an even number, the following formula for the roots may be useful: −b 2 ⎛b⎞ ± ⎜ ⎟ − ac x= 2 ⎝2⎠ . a 54 Example: Use the discriminant to determine the number of real solutions of the quadratic equation 3x 2 − 6 x + 3 = 0 Example: Use the quadratic formula to solve 3x + x 2 − 1 = 0 3 x 2 − 2 x − 10 = 0 55 The Quadratic Formula and Factoring: Theorem: If x1 and x2 are the roots of the quadratic equation ax 2 + bx + c = 0 , then the quadratic trinomial can be factored as follows: ax 2 + bx + c = a ( x − x1 )( x − x2 ) . Example: Factor by using the quadratic formula 3 4 x2 + 5x − 2 56 Applications Example: Find two consecutive positive integers such that the sum of their squares is 145. Geometry: Finding the Dimensions The height of a triangular sign is equal to its base. The area of the sign is 20 square feet. Find the base and the height of the sign. 57 Physics Using the position equation s = −16t 2 + υ0t + s0 , find the time when an object hits the ground if it is dropped from a building at a height of 320 feet. 58 Dimensions of a Field A farmer has 380 feet of fencing to enclose two adjacent fields. Find the dimensions that would enclose an area of 4000 square feet. 59 Constructing a Box An open box is formed by cutting 1.5 inch squares from each corner of a rectangular piece of metal whose length is twice its width and bending up the edges. If the box is to have a volume of 21 cubic inches, what dimensions should the piece of metal have? 60

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posted: | 8/2/2010 |

language: | English |

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