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CRASH COURSE IN QUADRATICS

Complete The Square. √. -b + b 2 – 4ac 2ac. (x+4)(x-3)=0. F O I L. (x+1)(x+2). X 2 – 5x +4. CRASH COURSE IN QUADRATICS. In preparation for the Algebra CST. Multiplying Polynomials. Area Model of Multiplication. 60 + 8. 30 + 4. To multiply 68 x 34:

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CRASH COURSE IN QUADRATICS

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  1. Complete The Square √ -b + b2 – 4ac 2ac (x+4)(x-3)=0 F O I L (x+1)(x+2) X2 – 5x +4 CRASH COURSE IN QUADRATICS In preparation for the Algebra CST

  2. Multiplying Polynomials Area Model of Multiplication 60 + 8 30 + 4 • 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. 1800+240+240+32=2312 Now you try one… 48 x 53

  3. Multiplying Polynomials Area Model of Multiplication x + 2 x + 3 • 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. X2 + + 6 5x Now you try one… (x+5)(x+1)

  4. Multiplying Polynomials FOIL ( x + 2 ) ( x + 3) = x2 + 5x + 6 (x)(x) = x2 First (x)(3) = 3x Outer (2)(x) = 2x Inner Last (2)(3) = 6 Combine like terms…

  5. Multiplying Polynomials x2 + 5x + 6 ax2 + bx + c a = 1 b = 5 c = 6

  6. Factoring Polynomials 10 6 14 12 2 5 6 1 7 2 3 4 7 7 9 7 21 18 6 3 5 4 9 10 Ask yourself… “What two numbers multiplied together give you the top digit and added together give you the bottom?”

  7. Factoring Polynomials X2 + 7x + 12 (x + )(x+ ) (x + )(x+ ) (x + )(x+ ) X2 + 13x + 36 -40 12 36 13 -6 7 X2 - 6x - 40

  8. Perfect Square Trinomial X2 + 12 + 36 (x + 6)(x + 6) (x + 6)2 X * X 6 * 6 X2 - 14 + 49 - (x - 7)(x - 7) (x - 7)2 X * X 7 * 7

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

  10. Graphing Quadratic Equations x2 – 4x = 0 0 02 – 4(0) 0 0, 0 2 -4 2, -4 22 - 4(2) 4 42 – 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 = 3 x = -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 -72 x2 - 21x = 72 x = x2 + 12x + 36 x = -21 • If x2 is added to x, the sum is 42. What are the values of x?

  14. Using Square Roots Square-Root Property x2 = 16 4x2 – 25 = 0 4x2 = 25 +25 +25 √4x2 = √25 √x2 = √16 2x = 5 x = +4 2 2 x2 = 16 x = + 2.5 (4)2= 16 (-4)2 = 16

  15. = 9 ( ) ( ) 6 2 6 2 2 2 ( ) b 2 2 Completing the Square Using Algebra Tiles x2 + 6x a= 1 b=6 c=0 x2 + 6x = 0 b = 6 + 9 + 9 x2 + 6x + 9 = 9 (x+3)(x+3)=9 (x+3)2 = 9 √(x+3)2 = √ 9 x+3= 3 x = 0 + x+3= 3 x+3= -3 x = -6

  16. Completing the Square Add to both sides of the equation ( ) 2 14 2 x2 + 14x = 15 b = 14 + 49 + 49 = 72 =49 Factor the Perfect Square x2 + 14x + 49 = 64 (x+7)(x+7)=64 (x+7)2 = 64 √(x+7)2 = √ 64 x+7= 8 x = 1 + x+7= 8 x+7= -8 x = -15

  17. Completing the Square Reduce 3x2 – 10x = -3 3 3 3 = 100 36 x2 - 10x = -1 3 b = 10 3 ( ) 2 Add to both sides of the equation -10 1 3 2 * x2 - 10x = -1 3 25 9 +25 9 +25 9 Factor the Perfect Square x2 - 10x + 25 = 16 3 9 9 -9 + 25 9 9 16 9 = 2 x – 5 = 4 3 3 x = 9 3 √ √( ) ( ) ( ) 2 x – 5 3 x – 5 3 x – 5 3 16 9 4 3 16 9 x = 1 3 = =+ = x – 5 = -4 3 3

  18. Completing the Square 2 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

  19. The Quadratic Formula a = 1 b = 5 c = 6 x2 + 5x + 6 ax2 + bx + c a = 2 b = 3 c = -5 2x2 + 3x – 5 = 0 ax2 + bx + c -b +√ b2 – 4ac 2a x = -b +√ b2 – 4ac 2a -3 +√ 49 4 -3 +√ 9 – (-40) 4 -3 +√ 32 – 4(2)(-5) 2(2) -3 +7 4 x = x = x = x = x = -3 + 7 4 -3 - 7 4 x = x = x = 4 x = - 2.5

  20. The Quadratic Formula 2x = x2 - 3 ax2 + bx + c 2x = x2 - 3 0 = x2 – 2x - 3 -2x -2x a = 1 b = -2 c = -3 0 = x2 – 2x - 3 ax2 + bx + c 2 +√ 16 2 2 - 4 2 2 +√ 16 2 -b +√ b2 – 4ac 2a 2 +4 2 2 + 4 2 -(-2) +√ (-2)2 – 4(1)(-3) 2(1) -b +√ b2 – 4ac 2a 2 +√ 4 +12 2 -(-2) +√ (-2)2 – 4(1)(-3) 2(1) x = x = x = x = x = x = x = x = x = x = x = 3 x = -1

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

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

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