1. CRASH COURSE IN QUADRATICS In preparation for the Algebra CST -b + b 2 – 4ac 2ac √ (x+4)(x-3)=0 (x+1)(x+2) X 2 – 5x +4 F O I L Complete The Square

2. Multiplying Polynomials Area Model of Multiplication (30)(60) 1800 (30)(8) 240 (4)(60) 240 (4)(8) 32 60+8 30 + 4 1800+240+240+32=2312 To multiply 68 x 34: Write the two numbers in expanded notation and multiply one box at a time. After you have multiplied the numbers, add all of the products together. Now you try one… 48 x 53

3. Multiplying Polynomials Area Model of Multiplication (x)(x) x 2 (x)(2) 2x (3)(x) 3x (3)(2) 6 x+2x+2 x+3x+3 X 2 + + 6 To multiply (x+2)(x+3): Write the two numbers in expanded notation and multiply one box at a time. After you have multiplied the numbers, add all of the products together. 5x Now you try one… (x+5)(x+1)

4. Multiplying Polynomials FOIL ( x + 2 ) ( x + 3) First (x)(x) = x 2 Outer (x)(3) = 3x Inner (2)(x) = 2x Last(2)(3) = 6 Combine like terms… = x 2 + 5x + 6

5. Multiplying Polynomials x 2 + 5x + 6 ax 2 + bx + c a = 1 b = 5 c = 6

6. Factoring Polynomials 34 12 7 25 10 7 61 6 7 72 14 9 35 6 4 18 9 21 10 Ask yourself… “What two numbers multiplied together give you the top digit and added together give you the bottom?”

7. Factoring Polynomials X 2 + 7x + 12 7 12 (x + )(x+ ) X 2 + 13x + 36 13 36 (x + )(x+ ) -6 -40 X 2 - 6x - 40

8. Perfect Square Trinomial X 2 + 12 + 36 X * X 6 * 6 (x + 6)(x + 6)(x + 6) 2 X 2 - 14 + 49 X * X 7 * 7 (x - 7)(x - 7)(x - 7) 2 -

9. Solving Quadratic Equations Graphing Factoring Using Square Roots Completing the Square Quadratic Formula

10. Graphing Quadratic Equations x 2 – 4x = 0 x y=x 2 - 4xyx, y 00 2 – 4(0)00, 0 2 2 2 - 4(2) -42, -4 44 2 – 4(4)0 4, 0 The Solution is the ________________

11. Find the solution for each graph:

12. Factoring Quadratic Equations Using the Zero Product Property (x-3)(x+7)=0 (x-3)=0(x+7)=0 x = 3x = -7

13. Factoring Quadratic Equations Solve using the Zero Product Property (x-3)(x+4)=0 (x+3)(2x-8)=0 (3x-1)(4x+1)=0 (3x+1)(8x-2)=0 Can you solve in your head? (x-2)(x+1)=0 x 2 + 12x + 36 x = x 2 - 21x = 72 x = -72 -21 If x 2 is added to x, the sum is 42. What are the values of x?

14. Using Square Roots Square-Root Property x 2 = 16 √ x 2 = √1 6 4x 2 – 25 = 0 x = +4 +25 √ 4x 2 = √ 25 2x = 5 2 x = + 2.5 4x 2 = 25 x 2 = 16 (4) 2 = 16 (-4) 2 = 16

15. Completing the Square Using Algebra Tilesx 2 + 6x a= 1 b=6 c=0 b2b2 ( ) 2 6262 2 x 2 + 6x = 0b = 6 + 9 x 2 + 6x + 9 = 9 (x+3)(x+3)=9 (x+3) 2 = 9 √ (x+3) 2 = √ 9 x+3 = 3 + x+3= -3 x = 0 x = -6 ( ) 6262 2 = 9

16. Completing the Square x 2 + 14x = 15b = 14 14 2 ( ) 2 = 7 2 =49 Add to both sides of the equation + 49 x 2 + 14x + 49 = 64 Factor the Perfect Square (x+7)(x+7)=64 (x+7) 2 = 64 √ (x+7) 2 = √ 64 x+7 = 8 + x+7= -8 x = 1 x = -15

17. Completing the Square x 2 - 10x = -1 3 b = 10 3 Add to both sides of the equation Factor the Perfect Square 3x 2 – 10x = -3 3 33 -10 1 3 2 ( ) 2 * = 100 36 Reduce 25 9 x 2 - 10x = -1 3 +25 9 +25 9 -9 + 25 9 9 = 16 9 x 2 - 10x + 25 = 16 3 9 9 x – 5 3 ( ) 4343 =+=+ x – 5 3 √ ( ) 2 16 9 = √ x – 5 = 4 3 3 x – 5 = -4 3 3 x = 9 3 x = 1 3 x – 5 3 ( ) 16 9 = 2

18. Completing the Square x - 8x = 12 x - 8x = 5 What should be added to both sides of this equation? x + 4x = 6 x - 4x = 8 ax – bx = c 2 2 2 2 2

19. The Quadratic Formula x 2 + 5x + 6ax 2 + bx + c a = 1 b = 5 c = 6 2x 2 + 3x – 5 = 0 ax 2 + bx + c a = 2 b = 3 c = -5 -b + √ b 2 – 4ac 2a x = -b + √ b 2 – 4ac 2a x = -3 + √ 3 2 – 4(2)(-5) 2(2) x = -3 + √ 9 – (-40) 4 x = -3 + √ 49 4 x = -3 + 7 4 x = -3 + 7 4 x = x = 4 -3 - 7 4 x = x = - 2.5

20. The Quadratic Formula -b + √ b 2 – 4ac 2a x = 2x = x 2 - 3ax 2 + bx + c2x = x 2 - 3 -2x 0 = x 2 – 2x - 3 ax 2 + bx + c a = 1 b = -2 c = -3 -(-2) + √ (-2) 2 – 4(1)(-3) 2(1) x = -(-2) + √ (-2) 2 – 4(1)(-3) 2(1) x = -b + √ b 2 – 4ac 2a x = 2 + √ 4 +12 2 x = 2 + √ 16 2 x = 2 + √ 16 2 x = 2 + 4 2 x = 2 + 4 2 x = x = 3 2 - 4 2 x = x = -1

21. Solving Quadratic Equations Graphing Factoring Using Square Roots Completing the Square Quadratic Formula

22. Solving Quadratic Equations x + 4x - 2 = 0 x - 5x + 4 = 0 2 2