Using the binomial square thm. rewrite the following trinomial into an equivalent expression (The square of a binomial) 25c 2 70cd + 49d 2

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Using the binomial square thm. rewrite the following trinomial into an equivalent expression (The square of a binomial) 25c 2 70cd + 49d 2 Dec 6 7:24 AM 1

Notes 6 1 Quadratic Expressions, Rectangles & Squares 1. What does the word quadratic refer to? 2. What is the general quadratic expression? 3. What is the general quadratic equation? Sum of possibly a constant, possibly a term with a variable to the first power, and definitely a term with a variable to the second power with no higher power ax 2 + bx + c, b and c can be any real number ax 2 + bx + c = 0 4. What is the general quadratic function? f(x) = ax 2 + bx + c 5. ax 2 + bx + c is a quadratic expression in what form? Standard Form 6. Give three different specific examples of a quadratic expression. 7. Is 2x + 5x 2 + 4 a quadratic expression? 2x 2 3x 2 + 7x 6 4x 2 + 4 x 2 + 5x 8. What is the simplest quadratic expression? Yes, this is just not in standard form x 2 9. The product of what two terms results in the simplest quadratic expression? The product of the simplest linear expressions x x = x 2 10. The product of two is a quadratic expression. 11. What types of formulas involve quadratic expressions? Linear expressions ax + b and cx + d Area formulas 12. Study example 1 13. Now try the following problem in your notes: a. Suppose a rectangular garden 25 feet by 15 feet has a walkway built around it. If the walkway is w feet wide, write the total area of the garden and walkway, in terms of w, in standard form. w w w 25 25 + 2w 15 15 + 2w w Area = (25 + 2w)(15 + 2w) 25 15 = (25 + 2w)15 + (25 + 2w)2w = 375 + 30w + 50w + 4w 2 = 4w 2 + 80w + 375 Jul 15 5:17 PM 2

Notes 6 1 Quadratic Expressions, Rectangles & Squares 14. When we square a linear expression, or take a linear expression to the 2 nd power, or multiply a linear expression by itself the result is: (x + 2) 2 = (x + 2)(x + 2) = x 2 + 4x + 4 15. The processes of squaring a linear expression is called: A quadratic expression Expanding the power 16. Study example 2 17. Now try the following problem in your notes: a. (2x 5y) 2 = (2x 5y)(2x 5y) = (2x 5y)(2x) + (2x 5y)( 5y) = 4x 2 10xy + 10xy 25y 2 = 4x 2 20xy + 25y 2 Jul 15 5:17 PM 3

Notes 6 1 Quadratic Expressions, Rectangles & Squares 18. What is one way to think about squaring a binomial that can help you organize the process? The area of a square whose side is the binomial 19. Study example 3 20. Now try the following problem in your notes: a. Write the area of the square with sides of length a + b in standard form. Draw a picture of the square. a + b a + b a + b a b a b a + b = (a+b) 2 =(a+b)(a+b) = a 2 +2ab+b 2 Jul 15 5:17 PM 4

Notes 6 1 Quadratic Expressions, Rectangles & Squares 21. Binomial Square Theorem must memorize 22. Study example 4 23. Now try the following problem in your notes: a. = (2w) 2 2(2w)(1/2) + (1/2) 2 = 4w 2 2w + 1/4 Jul 15 5:17 PM 5

Notes 6 1 Quadratic Expressions, Rectangles & Squares Quadratic Expressions from Rectangles A 16 x 24 picture is surrounded by a border that is x inches wide. Find the information shown below. x Outside dimensions of the picture and the border: x 16" x 24" x Total area of picture and border: Area of border only: Jul 15 5:17 PM 6

Find the area of the walkway x+9 3x 3x + 5 2(x+6) Jul 15 5:43 PM 7

Quadratic Expressions From Squares 1. ( x + 8) 2 Shortcut: ( x + 8) 2 2. ( 3x 5) 2 3. 9 ( x 2 ) 2 Jul 15 6:08 PM 8

Expand & Simplify 4. 1/3( 3d + 6 ) 2 5. ( 2x 1 ) 2 ( x + 5 ) 2 6. ( 5x 7y ) 2 7. ( x 2 ) 2 + ( 2x 1) 2 Jul 15 6:18 PM 9

Solve for h 8. x 2 6x + 9 = ( x + h ) 2 9. x 2 + 14x + 49 = ( x + h ) 2 Jul 15 6:24 PM 10

Binomial Square Theorem ( x + y ) 2 = x 2 + 2xy + y 2 ( x y ) 2 = x 2 2xy + y 2 Sep 13 8:42 AM 11

Expand each. ( 1/2 a + 12b ) 2 3 ( x y ) 2 Sep 13 8:45 AM 12

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(7e) 2 + 2(7e)(3f) + (3f) 2 7e + 3f 49e 2 + 42ef + 9f 2 49e 2 21ef 21ef 9f 2 7e + 3f 49e 2 + 42ef + 9f 2 Nov 30 7:25 AM 21

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OR (x + 25) 2 = (x + h) 2 Nov 30 7:11 AM 27

OR 1 = h 1 = h Nov 30 7:26 AM 28

AdvAlg6.1QuadraticExpressionsRectanglesSquares.notebook Nov 23 4:07 PM 29

AdvAlg6.1QuadraticExpressionsRectanglesSquares.notebook Nov 23 4:07 PM 30

AdvAlg6.1QuadraticExpressionsRectanglesSquares.notebook Nov 23 4:07 PM 31

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