Unit 2 Quadratic Functions and Factoring

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Unit 2 Quadratic Functions and Factoring by Mind Map: Unit 2 Quadratic Functions and Factoring

1. Ex: (3+2i)/(4+5i)× (4- 5i)/(4- 5i)=21/24

2. Multiplying

2.1. Ex: (3+2i)(4+5i) 12+15i+8i+10i^2 12+23i- 10 2+23i

3. Adding and Subtracting

3.1. Ex: (3+2i)-(4+5i) - 1- 3i

4. Factor by grouping (four terms)

4.1. 1) split the four terms into two groups 2) find the greatest common factor or shared factor of each group

4.2. Ex: 8r^3- 64r^2+r- 8 (8r^3- 64r^2)+(r- 8) 8r^2 (r- 8)+1 (r- 8) (8r^2+1)(r- 8)

5. Patterns in factoring

5.1. Difference of squares Ex: x^2=9

5.2. Perfect square trinomials Ex: x^2- 10x+25

5.3. Sum of cubes Ex: x^3+125 (a+b)(a^2- ab+b^2) Same Opposite Always Positive!

5.3.1. Ex: x^3+4^3 (x+4)(x^2- 4x+16) Check work: x^3- 4x^2+16x+4x^2- 16x+64=x^3+64

5.4. Difference of cubes Ex: x^3- 125 (a- b)(a^2+ab+b^2) Same Opposite Always Positive!

5.4.1. Ex: x^3- 3^3 (x- 3)(x^2+3x+9) Check work: x^3+3x^2+9x- 3x^2- 9x- 27=x^3- 27

6. Factoring trinomials

6.1. With leading coefficient= 1

6.1.1. Ex: x^2- 3x- 28 - 7×4=- 28 - 7+4=- 3 (x- 7)(x+4)

6.2. With leading coefficient = not 1

6.2.1. Ex: 6x^2- 11x- 2 x^2- 11x- 12 (6x^2- 12x)+(x- 2) 6x (x- 2)+1 (x- 2) (6x+1)(x- 2)

7. Solving Quadratic Equations

7.1. With leading coefficient= 1

7.1.1. 1) Variables on the left, constants on the right. 2) Take 1/2 of the coefficient of the "x" variable term. 3) Square that value and add to both sides of the equation. 4) This makes the left side a Perfect Square Binomial.

7.1.1.1. Ex: x^2+8x- 43=0 x^2+8x=- 43 8/2=4 x^2+8x+16=43+16 x^2+8x+16=59 (x+4)^2=59

7.2. With leading coefficient= not 1

7.2.1. 1) Variables on the left, constants on the right. 2) Make leading coefficient =1 by dividing every term by the leading coefficient. 3) Take 1/2 of the coefficient of the "x" variable term. 4) Square that value and add to both sides of the equation. 5) This makes the left side a Perfect Square Binomial.

7.2.1.1. Ex: 3x^2- 6x=31 x^2- 2x=31/3 - 2/2=- 1 x^2- 2x+1=31/3+3/3 (x- 1)^2=34/3

8. Discriminants

8.1. (b)^2- 4 (a)(c)

8.2. D>0: 2 x- intercepts 2 solutions

8.3. D=0 1 x- intercept 1 solution

8.3.1. D<0 no x- intercepts 2 solutions

9. Imaginary number patterns

9.1. i^1=i

9.2. i^2= - 1

9.3. i^3= - i

9.4. i^4= 1

10. Dividing imaginary numbers