notes packet on quadratic functions and factoring · notes packet on quadratic functions and...

Notes Packet on Quadratic Functions and Factoring Graphing quadratic equations in standard form, vertex form, and intercept form.

A. Intro to Graphs of Quadratic Equations: ! • A ____________________ is a function that can be written in the form !

where a, b, and c are real numbers and a! 0. • The graph of a quadratic function is a U-shaped curve called a ________________.

The maximum or minimum point is called the _____________ Identify the vertex of each graph; identify whether it is a minimum or a maximum. 1.) 2.)

Vertex: ( , ) _________ Vertex: ( , ) _________

3.) 4.)

Vertex: ( , ) _________ Vertex: ( , ) _________

B. Key Features of a Parabola:

Without graphing the quadratic functions, complete the requested information:

2y ax bx c= + +2y ax bx c= + +

≠

5.) !

What is the direction of opening? _______ Is the vertex a max or min? _______ Wider or narrower than y = x2 ? __________

2( ) 3 7 1f x x x= − +

8.) !

What is the direction of opening? _______ Is the vertex a max or min? _______ Wider or narrower than y = x2 ? ___________

20.6 4.3 9.1y x x= − + −

7.) !

What is the direction of opening? _______ Is the vertex a max or min? _______ Wider or narrower than y = x2 ? __________

22 113

y x= −

6.) ! What is the direction of opening? _______ Is the vertex a max or min? _______ Wider or narrower than y = x2 ? ___________

25( ) 34

g x x x= − + −

C. Graphing in STANDARD FORM ( ! ): we need to find the vertex first. Find the vertex of each parabola. Graph the function and find the requested information

The parabola y = x2 is graphed to the right.

Note its vertex (___, ___) and its width.

You will be asked to compare other parabolas to this graph.

!

2y ax bx c= + +

Vertex: _______ Max or min? _______ Direction of opening? _______ Axis of symmetry: ________ Compare to the graph of y = x2

_________________________


_________________________

9.) f(x)= -x2 + 2x + 3 a = ____, b = ____, c = ____

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

10.) h(x) = 2x2 + 4x + 1

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

12.) State whether the function y = −3x2 + 12x − 6 has a minimum value or a maximum value. Then find the minimum or maximum value.

13.) Find the vertex of ! . State whether it is a minimum or maximum. Find that minimum or maximum value.


_________________________

11.) k(x) = 2 – x – x2

!

12

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

21 5 72

y x x= + −

Find the vertex of each parabola and graph.

15.) Write a quadratic function in vertex form for the function whose graph has its vertex at (-5, 4) and passes through the point (7, 1).

Another useful form of the quadratic function is the vertex form: ________________________________.

GRAPH OF VERTEX FORM y = a(x − h)2 + k The graph of y = a(x − h)2 + k is the parabola y = ax2 translated ___________ h units and ___________ k units. • The vertex is (___, ___). • The axis of symmetry is x = ___. • The graph opens up if a ___ 0 and down if a ___ 0.

Vertex: _______

Vertex: _______

13.) !

!

( )21 2y x= + −

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

14.) !

!

( )21 1 53

y x= − − +

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

Converting between forms:

GRAPH OF INTERCEPT FORM y = a(x − p)(x − q): Characteristics of the graph y = a(x − p)(x − q): • The x-intercepts are ___ and ___.

• The axis of symmetry is halfway between (___, 0) and ( ___ , 0)

and it has equation x =!

• The graph opens up if a ___ 0 and opens down if a ___ 0.

2

x-intercepts: _______, _______

Vertex: _______

16.) Graph y = −2(x − 1)(x − 5)

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

From intercept form to standard form • Use FOIL to multiply the binomials

together • Distribute the coefficient to all 3 terms

From vertex form to standard form • Re-write the squared term as the

product of two binomials • Use FOIL to multiply the binomials

together • Distribute the coefficient to all 3 terms • Add constant at the end

Ex: ! ( )( )2 5 8y x x= − + −

Ex: ! ( ) ( )24 1 9f x x= − +

Notes 16: Solving quadratics by Factoring

A. Factoring Quadratics

Strategies to use: (1) Look for a GCF to factor out of all terms (2) Look for special factoring patterns as listed below (3) Use the X-Box method (4) Check your factoring by using multiplication/FOIL

! Factor each expression completely. Check using multiplication.

Examples of monomials:_______________________________

Examples of binomials:________________________________

Examples of trinomials:________________________________

1.) !23 15x x−

5.) !2 22 121m m− +

3.) ! 2 5 24x x− −

4.) !225 81x −

2.) !26 24x −

6.) ! 24 12 9x x+ +

B. Solving quadratics using factoring To solve a quadratic equation is to find the x values for which the function is equal to _____. The solutions are called the _____ or _______of the equation. To do this, we use the Zero Product Property:

Zero Product Property List some pairs of numbers that multiply to zero:

(___)(___) = 0 (___)(___) = 0 (___)(___) = 0 (___)(___) = 0

What did you notice? _______________________________________________

9.) 25t2 − 110t + 121 7.) ! 25 17 6x x− +

11.) ! 29 42 49a a+ +10.) !

216 36x −

8.) ! 23 5 12x x+ −

12.) ! 26 33 36x x+ +

ZERO PRODUCT PROPERTY

If the _________ of two expressions is zero, then _______ or _______ of the expressions equals zero. Algebra If A and B are expressions and AB = ____ , then A = _____ or B = __. Example If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is, x = __________ or x = _________.

Use this pattern to solve for the variable: 1. get the quadratic = 0 and factor completely 2. set each ( ) = 0 (this means to write two new equations) 3. solve for the variable (you sometimes get more than 1 solution)

Find the roots of each equation:

Find the zeros of the function by rewriting the function in intercept form:

Find the zeros of each equation:

17.) v(v + 3) = 10

13.) ! 2 7 30 0x x+ − =

18.) 22 15x x+ =

15.) 23 2 21x x− =

14.) !

2 6 4 8 03 7 5 9x x⎛ ⎞⎛ ⎞− + =⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

16.) !22 8 30 34x x x+ − = −

19.) ! 26 3 63y x x= − − 20.) ! ( )

212 6 6f x x x= + − 21.) ! ( ) 249 16g x x= −

Graph the function. Label the vertex and axis of symmetry:

Vertex: _______ Maximum or minimum value: _______ x-intercepts: _______ Axis of symmetry: ________ Compare width to the graph of y = x2

_________________________

22.) !

( )2 1 ( 3)y x x= − − +

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

Notes #17: Solve Quadratic Equations by Finding Square Roots

A. Simplifying Square Roots: • Make a factor tree; circle pairs of “buddies.” • One of each pair comes out of the root, the non-paired numbers stay in the root. • Multiply the terms on the outside together; multiply the terms on the inside together

Simplify:

B. Multiplying Square Roots: • Simplify each radical completely by taking out “buddies”

• (outside • outside)! or ! • Simplify your answer, if possible

Simplify:

C. Simplifying Square Roots in Fractions:

• Split up the fraction: ! • Simplify first by taking out “buddies” or reducing (you can only reduce two numbers

that are both under a root or two numbers that are both not in a root) • Square root top, square root bottom • If one square root is left in the denominator, multiply the top and the bottom by the

square root and simplify OR If a binomial is left in the denominator, then multiply top and bottom by the conjugate of the denominator (exact same expression except with the opposite sign). Remember to FOIL on the denominator.

• Reduce if possible Simplify:

1.) !3 120 2.) ! 5 72−

inside inside� ( )( ) ________a b c d =

3.) !( )( )5 12 3 154.) ! ( )( )2 5 2 5

2 2 2 25 5 25

ab= = =

5.) !

73 6.) !

45+ 3 7.) !

2 53 2−

+

D. Solving Quadratic Equations Using Square Roots

• Isolate the variable or expression being squared (get it ______________) • Square root both sides of the equation (include + and – on the right side!) • This means you have _____________ equations to solve!! • Solve for the variable (make sure there are no roots in the denominator)

8.) x2 = 25 9.) 3x2 = 81

10.) 4x2 – 1 = 0 11.)

12.) (2y + 3)2 = 49 13.)

2

3 215m

− =

( )23 2 6 34x − − =

Notes#18: Complex Numbers and Completing the square Complex Numbers A. Definitions

B. Solving a quadratic equation with complex roots • Isolate the expression being squared • Square root both sides; write two equations Replace ! with i. Simplify

Solve

C. Adding, subtracting, and multiplying complex numbers • Distribute/FOIL. Combine like terms. • Replace ! with (-1). Simplify.

Simplify

Define Complex Numbers: imaginary unit (i): imaginary number :

1−

1.) x2 = −27 2.) 2x2 + 11 = −37 3.) ! 24(2 1) 8 0x − + =

2i

4.) (3 + 7i) − (8 − 2i) 5.) (2 + 5i) + (7 − 2i) 6.) (2 + i)(−5 + 2i) 7.) (3 − i)(5 − 2i)

B. Dividing complex numbers • If i is part of a monomial on the denominator, multiply top and bottom by i. ex:

! • If i is part of a binomial on the denominator, multiply top and bottom by the

complex conjugate of the denominator (same expression but opposite sign).

FOIL. ex: ! • Replace ! with (-1). Simplify.

56i

56 i−

2i

8.) !34i 11.) !

7 51 4

ii

+

−10.) !27ii−

9.) !6 42

ii

+

+

Completing the Square

B. Review: Solving Using Square Roots • Factor and write one side of the equation as the square of a binomial • Square root both sides of the equation (include + and – on the right side; 2 equations! • Solve for the variable (make sure there are no roots in the denominator)

1) (k + 2)2 = 12 2.) x2 + 2x + 1 = 8 3.) n2 – 14n + 49 = 3

C. Completing the Square ! • Take half the b (the x coefficient) • Square this number (no decimals – leave as a fraction!) • Add this number to the expression • Factor – it should be a binomial, squared ( )2

4.) x2 + 6x + _____ 5.) m2 – 14m + _______ ( )( )

( )2

Find the value of c such that each expression is a perfect square trinomial. Then write the expression as the square of a binomial. 6.) w2 + 7w + c 7.) k2 – 5k + c

8.) x2 + 4x – 5 = 0 9.) m2 – 5m + 11 = 10

2 0ax bx c+ + =

Solving by Completing the Square: • Collect variables on the left, numbers on the right • Divide ALL terms by a; leave as fractions (no decimals!) • Complete the square on the left – add this number to BOTH sides • Square root both sides (include a ______ and _______ equation!) • Solve for the variable (simplify all roots – look for ! ) 1 i− =

10.) ! 11.) !

12.) 2x2 – 3x – 1 = 0 13.)

Cumulative Review: Solving Quadratics

Solve by factoring: 14.) 12k2 – 5k = 2 15.) 49m2 – 16 = 0

Solve by using square roots: 16.) 4w2 = 18 17.) 3y2 – 8 = 0

22 9 3 17k k k+ − = − 23 4 11 2 2w w w+ + = −

23 2 6x x x+ = −

Notes #19: Use the Quadratic Formula and the Discriminant A. Review of Simplifying Radicals and Fractions • Simplify expression under the radical sign ( ! ); reduce • Reduce only from ALL terms of the fraction. • (You can’t reduce a number outside of a radical with a number inside of a radical) • Make sure that you have TWO answers

Simplify:

1 i− =

1.) ! 6 182

− ±

4.) ! 8 272

− ±

3.) ! 4 20

4± −

2.) ! 5 202

− ±

6.) !

29 (6) 4( 3)( 3)4

− ± − − −

5.) !

29 ( 5) (5)(2)(3)4

− ± − −

B. Solving Quadratics using the Quadratic Formula

So far, we have solved quadratics by: (1) _______________, (2) ______________, and (3) ___________________.

The final method for solving quadratics is to use the quadratic formula.

Solve by using the quadratic formula:

1.) x2 + x = 12

(std. form):

a = _____

b = _____

c = _____

2.) 5x2 – 8x = -3

(std. form):

a = _____

b = _____

c = _____

Solving using the quadratic formula: • Put into standard form (ax2 + bx + c = 0) • List a = , b = , c =

• Plug a, b, and c into ! • Simplify all roots (look for ! ); reduce

2 42

b b acx

a− ± −

=

1 i− =

2 42

b b acx

a− ± −

=

2 42

b b acx

a− ± −

=

3.) -x2 + x = -1 4.) 3x2 = 7 – 2x

5.) -x2 + 4x = 5 6.)

24( 1) 6 2x x− = +

C. Using the Discriminant • Quadratic equations can have two, one, or no solutions (x-intercepts). You can determine

how many solutions a quadratic equation has before you solve it by using the ________________.

• The discriminant is the expression under the radical in the quadratic formula:

!

A. Finding the number of x-intercepts Determine whether the graphs intersect the x-axis in zero, one, or two points.

1.) ! 2.) !

B. Finding the number and type of solutions Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. 3.) ! 4.) !

5.) 9x2 – 6x = 1 6.) 4x2 = 5x + 3

2 42

b b acx

a− ± −

=

Discriminant = b2 – 4ac

If b2 – 4ac < 0, then the equation has 2 imaginary solutions

If b2 – 4ac = 0, then the equation has 1 real solution

If b2 – 4ac > 0, then the equation has 2 real solutions

24 12 9y x x= − + 23 13 10y x x= − −

23 5 1x x− = 2 3 7x x= − −

Cumulative Review Problems: Solve by factoring: 7.) 4m2 +5m – 6 = 0 8.) 3x3 – 27x = 0

Solve by using square roots: 9.) 4b2 + 1 = 0 10.) (3x + 1)2 = 18

Solve by completing the square: 11.) 4m2 + 12m +5 = 0 12.) x2 – 7x – 18 = 0

For #13, find the vertex of the parabola. Graph the function and find the requested information

Vertex: _______

Max or min value: _______

Direction of opening? _______

Compare width to y = x2 :

_______________

Axis of symmetry: ________

13.) g(x) = -2x2 + 8x – 5

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

Notes #20: Graph and Solve Quadratic Inequalities

GRAPHING A QUADRATIC INEQUALITY IN TWO VARIABLES

To graph a quadratic inequality, follow these steps:

Step 1 Graph the parabola with equation y = ax2+ bx + c. Make the parabola _______________ for inequalities with < or > and ______________ for inequalities with ≤ or ≥.

Step 2 Test a point (x, y) ______________ the parabola to determine whether the point is a solution of the inequality.

Step 3 Shade the region ______________ the parabola if the point from Step 2 is a solution. Shade the region ________________ the parabola if it is not a solution.

1.) y ≤ −x2 + 2x + 3

2.) y ≥ x2 + 3x – 4

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

3.) y > 3x2 +3x− 5 y < −x2 +5x + 10

4.) y < -x2 +4 y > x2 − 2x− 3

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

Notes# 21: Write Quadratic Functions and Models

A. When given the vertex and a point • Plug the vertex in for (h, k) in ! • Plug in the given point for (x, y) • Solve for a. Plug in a, h, k into !

1.) Write a quadratic equation in vertex form for the parabola shown.

2.) Write a quadratic function in vertex form for the function whose graph has its vertex at (2, 1) and passes through the point (4, 5).

B. When given the x-intercepts and a third point • Plug in the x-intercepts as p and q into y = a(x − p)(x − q) • Plug in the given point for (x, y) • Solve for a. Plug in a, h, k into y = a(x − p)(x − q)

3.) Write a quadratic function in intercept form for the parabola shown.

2( )y a x h k= − +

2( )y a x h k= − +

B. When given three points on the parabola • Label all three points as (x, y) • Separately, plug in each point into ! • You now have 3 equations with three variables: a, b, c • Solve for a, b, and c using elimination (see notes #13). Plug back into

!

4.) Write a quadratic function in standard form for the parabola that passes through the points (−2, −6), (0, 6) and (2, 2).

5.) Write a quadratic function in standard form for the parabola that passes through the points (−1, −2), (1, −4) and (2, 1).

2y ax bx c= + +

2y ax bx c= + +

Notes #22: Review

To graph a quadratic function, you must FIRST find the vertex (h, k)!!

(A) If the function starts in standard form ! :

1st: The x-coordinate of the vertex, h, = 2nd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function & solving for y.

(B) If the function starts in intercept form ! : 1st: Find the x-intercepts by setting the factors with x equal to 0 & solving for x. 2nd: The x-coordinate of the vertex is half way between the x-intercepts. 3rd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function & solving for y.

(C) If the function starts in vertex form ! : 1st: pick out the x-coordinate of the vertex, h. REMEMBER: h will have the OPPOSITE sign as what is in the parenthesis!! 2nd: Pick out the y-coordinate of the vertex, k. It will have the SAME sign as the what is in the equation!

AFTER finding the vertex: Make a table of values with 5 points: The vertex, plug in 2 x-coordinates SMALLER than the x-coordinate of the vertex & 2 x-coordinates LARGER than the x-coordinate of the vertex.

Direction of Opening: If a is positive, the graph opens up If a is negative, the graph opens down.

Width of the function:

If ! , the graph is narrower than!

If ! , the graph is wider than !

2y ax bx c= + +

2ba−

( )( )y a x p x q= − −

2( )y a x h k= − +

1a > 2y x=

1a < 2y x=

Graph each function by making a table of values with at least 5 points. (A) State the vertex. (B) State the direction of opening (up/down). (C) State whether the graph is

wider, narrower, or the same width as ! . 2y x=

Vertex: _______ Direction of opening? _______ Compare width to the graph of y = x2

_________________________


_________________________


_________________________

1.) !

!

21( ) ( 6) 52

f x x= + −

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

2.) k(x) = x2 + 2x + 1

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

3.) f(x) = x2 – x – 6

!

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y


_________________________


_________________________


_________________________

4.) !

!

2( ) 2 3f x x x= − + +

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

5.) !

!

2( ) 2 4 1h x x x= + +

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

6.) !

!

1( ) ( 4)( 6)3

g x x x= − − +

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10x

-10-9-8-7-6-5-4-3-2-1

12345678910y

Methods for Solving Quadratic Equations: A.) Factoring 1st: Set equal to 0 2nd: Factor out the GCF 3rd: Complete the X & box method to find the factors 4th: Set every factor that contains an x in it, equal to 0 & solve for x.

B.) Completing the Square 1st: Move the constant (number with no variable) to the right so that all variables are on the left& all constants are on the right. 2nd: Divide every term in the equation by the value of a, if it is not already 1.

3rd: Create a perfect square trinomial on the left side by adding to both sides.

4th: Factor the left side into a and simplify the value on the right side. 5th: Take the square root of both sides of the equation. REMINDER: Don’t forget the 6th: Solve for x

C.) Finding Square Roots 1st: Isolate the term with the square. 2nd: Take the square root of both sides of the equation. REMINDER: Don’t forget the 3rd: Solve for x.

D.) Quadratic Formula 1st: Set the equation equal to 0. 2nd: Find the values of a, b, and c & plug them into the Quadratic Formula:

! 3rd: Simplify the radical as much as possible. 4th: If possible, simplify the numerator into integers.

5th: Divide. REMINDER: If you have 2 terms in the numerator (ex: ), divide BOTH terms by the number in the denominator (the example would result in

! )

Examples: Solve each equation by the method stated.

2

2b⎛ ⎞⎜ ⎟⎝ ⎠

2

2b

x⎛ ⎞±⎜ ⎟⎝ ⎠

±

±

2 42

b b acx

a− ± −

=

4 6 32

±

2 3 3±

By Square Roots:

1.) (2y + 3) 2 = 49 2.) (3m – 1) 2 = 20 3.) !2(3 5) 1 11r + − =

By Factoring:

4.) x2 – 4x – 5 = 0

By Completing the Square:

7.) x2 – 6x – 11 = 0 8.) 2y2 + 6y – 18 = 0 9.) 2x2 – 3x – 1 = 0

By Quadratic Formula:

10.) x2 + x = 12 11.) 5x2 – 8x = -3 12.) 2x2 = 4 – 7x

6.) 22 15x x+ =5.)

23 2 21x x− =

Chapter 4 Review Sheet

Please complete each problem on a separate sheet of paper. Show all of your work and please use graph paper for all graphs.

For questions 1-3, solve by factoring.

1. ! 2. ! 3. !

For questions 4-5, solve by finding square roots.

4. ! 5. !

For questions 6-8, solve by completing the square.

6. ! 7. ! 8. !

For questions 9-10, solve by using the quadratic formula. 9. ! 10. !

For questions 11-13, simplify each expression.

11. ! 12. ! 13. !

For questions 14-16, write each function in vertex form, graph the function, and label the vertex and axis of symmetry.

14. ! 15. ! 16. !

For question 17-19, graph each inequality.

17. ! 18. ! 19. !

For questions 20-21, write a quadratic function in vertex form whose graph has the given vertex and passes through the given point. 20. Vertex (2, 3) and passes through (-3, 7) 21. Vertex (-1, -5) and passes through (2, -1)

For questions 22-23, write a quadratic function in standard form whose graph passes through the given points. 22. (1, 1), (0, -2), (2, 8) 23. (-1, -7), (1, -5), (2, -1)

For questions 24-28, write the expression as a complex number in standard form. 24. ! 25. ! 26. !

27. ! 28. !

2 12 32 0x x− + = 23 8 5 0x x− + = 22 4 30 0x x+ − =

21 ( 1) 16 04x + − = ( )22 1 3 6x− − + =

2 16 15 0x x− + = 25 10 20 0x x− + + =21 4 6 0

2x x− + − =

2 3 5 0x x− − − = 2 23 2 5 1x x x x+ = + −

48 324± 5

48 252 3⋅

2 16 2y x x= − + 22 4 7y x x= − − +21 ( 4) 8

3y x= − −

22( 1) 5y x> + − 2 8 2y x x≤ − + + 22 12 16y x x≤ − +

(7 3 ) (9 4 )i i− + + (6 2 )( 8 3 )i i− − + ( 3 7 ) (9 2 )i i− − − −

2 57

ii

+

−

4 73ii−

notes packet on quadratic functions and factoring · notes packet on quadratic functions and...

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