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U n i t 3 P t 1 H o n o r s P a g e | 1

Math 3 Unit 3 Day 1 - Factoring Review

I. Greatest Common Factor – GCF

Examples:

A. 23 6x x B. 24 8 4x x C. 2 2 216 8x y x

II. Difference of Two Squares

Draw ( - ) ( + )

Square Root 1st and Last Term

Examples:

A. 28 18x B. 23 27x C. 416 81x

III. Grouping

Group 1st two terms and last two terms with parentheses

Take out a GCF from first and last grouping (Hint: your parentheses should be the same)

Take out a GCF (the common parentheses) from that step

A. 3 22 4 8x x x B. 3 24 6 10 15x x x C. 2 2 2 2ax bx ay by

IV. Trinomials with Leading Coefficient of 1

E. 2 7 12x x F. 2 5 36x x

G. 2 10 16x x H. 2 2 24x x


V. Trinomials with Leading Coefficient Greater than 1.

A. 26 21 9x x B. 22 5 3x x

C. 26 22 12x x D. 24 5 6x x

VI. Sum or Difference of Two Cubes

Patterns: a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2)

Steps:

Draw - ( ) ( )

Signs - Same, Opposite, Always Positive

To Get Little ( ) - Cube root each term

To Get Big ( ) - Square 1st, Multiply Middle, Square Last

Factor the following.

1. 3 27x 2. 38 64x

3. 32 250x 4. 3216 1x

5. 6 3a b

6. 3 3125 64a b


Math 3 Unit 3 Day 2 - Solving Quadratic Equations by Factoring and Imaginary Numbers Part 1

Quadratic Equations The graph is a parabola, a u-shaped figure.

Parts of a Parabola:

Vertex

Line of Symmetry/Axis of Symmetry

Direction of opening

X and Y intercepts

Solutions: where the graph touches/crosses the ____________________.

*Terms used for solutions of quadratic equations: ___________, ____________, ___________.

*There are three possible outcomes when solving quadratic functions:

The Fundamental Theorem of Algebra:

The degree of a polynomial (the highest exponent) indicates____________________________

I. Solving Quadratic Equations

Steps:

1. Set the equation equal to zero (move everything to one side).

2. Factor the polynomial.

3. Set each factor equal to zero and solve.

*The number of solutions ________________________________________________.

Examples:

1. 2 7 12x x 2. 23 48x


3. 3 22 2 0x x x 4. 4 210 40 0x x

5. 23 24 48 0x x 6. 22 7 15x x

II. Write the quadratic equation given the roots.

1. -5 , 6 2. 1

, 23

3. 2 3

,3 4

4.

2 1,

5 2

III. Complex Numbers

Example: What is the square root of -25?


If 1i , what is 2i ?

Which way is correct?

1i

1i

Rules:

1.

2.

Examples:

1. 9 2. 36

3. 2(3 )i 4.

2( 5)i

5. 2( 2) 6. 9i

7. 6i 8. 15i

9. 99i 10. 25i


Math 3 Unit 3 Day 3 – Imaginary Numbers Part 2, Completing the Square and Quadratic Formula

I. Operations with Imaginary Numbers

1. 6 (3 8)i i 2. 3 4 (5 8 ) 2(5 9)i i i

3.. (12 7 )(10 9 )i i 4.. 2( 2)i

II. Equations

1. 25 45 0x 2. 24 24 0x

III. Find the value of each variable.

1. 2 3 ( 4) 3 2m n i i 2. (3 4) (3 ) 16 3m n i i

3. ( ) (7 2 ) 9 9m n i m n i 4. (10 9 ) (5 4 ) 15 10m n m n i i


IV. Completing the Square

A. Write the trinomial as the square of a binomial.

1. 2 2 1x x 2. 2 10 25x x 3. 2 12 36x x

B. Find the value of “C” that makes the trinomial a perfect square.

1. 2 6 " "x x C 2. 2 8 " "x x C 3. 2 4 " "x x C

C. Square Root Property

1. 2( 5) 7x 2. 21( 3) 8

4x

3. 2 14 49 64x x 4. 29 6 1 2x x

V. Completing the Square

A. Steps: 1. Isolate the constant (move the constant to the other side).

2. Make sure the leading coefficient is 1 (if not divide through).

3. Complete the Square (Divide the middle number by 2 and square it).

4. Add that number to both sides of the equation.

5. Factor the left side and combine the right side (Short cut for factoring - Square root the first term, square

root last term, and take the first sign).

6. Square root both sides of the equation. (Remember to put a ± on the right.)

7. Solve the equation.

YOU SHOULD HAVE TWO ANSWERS FOR EACH PROBLEM.


Examples:

1. 2 6 4 0x x 2. 23 12 2 0x x

3. 24 8 1 0x x 4. 24 2 5 0x x 5. 23 2 8 0x x

VI. Quadratic Formula 2 4

2

b b ac

a

Put the equation in standard form to find a, b, and c.

Examples:

1. 23 5 6x x 2. 2 6 9x x

3. 23 6 4x x 4. 23 5 2x x


Math 3 Unit 3 Day 4 - Graphing Parabolas and Circles

I. Parabolas -

Up/Down 2( )y a x h k

a = Positive opens up

a = negative opens down

Vertex (h,k)

Over Up or Down

1

2

3

Steps if Equation for Parabola is not in Standard Form (Similar to Completing the Square)

1. Move the constant to the other side.

2. Make sure the leading coefficient is 1. If not, divide through.

3. Complete the square. Remember to balance the equation and add that number to both sides.

4. Factor the trinomial you created and combine like terms on the other side.

5. Solve for y.

Decide the direction the parabola opens; write the equation in standard form, state the vertex, and

graph.

1. 22 4 6y x x 2. 22 8 6y x x

Opens___________ Opens____________

Vertex_________________ Vertex_______________

Examples:

Decide if the parabola opens up or down; state the vertex, and

graph. 22( 1) 4y x

Vertex______________


3. 22 16 124y x x 4. 23 12 13y x x

Opens___________ Opens____________

Vertex_________________ Vertex_______________

II. Circles

Write the equation of each circle in standard form and graph.

Steps: 1. Group all the “like terms” together. 2. Create two perfect square trinomials by completing the square AND balance the equation. 3. Factor both trinomials. You now have the center and the radius.

1. 9𝑥2 + 54𝑥 + 9𝑦2 − 18𝑦 + 64 = 0


2. 𝟒𝒙𝟐 − 𝟒𝒙 + 𝟒𝒚𝟐 − 𝟓𝟗 = 𝟎 3. 𝟐𝒙𝟐 + 𝟐𝒚𝟐 − 𝟏𝟎𝒙 + 𝟐𝒚 − 𝟓 = 𝟎


Math 3 Unit 3 Day 5 Parabolas…..graphing in a new way with Focus and Directrix

Understanding the Pieces

Graph the point (3,1).

Go exactly 2 units up from the vertex and

place a point.

Go exactly 2 units down from the vertex

and place a point then draw a horizontal line

through that point.

Practice 1:

Focus is (5,-3)

Directrix is y = 3

What is the vertex?

Which way does it open?


Practice 2:

Directrix is y = 2

Vertex is (-1,1)

What is the focus?


Practice 3:

Vertex is (-2,-4)

Focus is (-2,-2)

What is the directrix?


Practice 4:

Vertex is (4,6)

Focus is (4,8)

The length of the latus rectum is 8.

Graoh the parabola.

Practice 5:

Vertex is (10,-6)

Focus is (10,-2)

The length of the latus rectum is 6.

Graph the parabola.

1. Graph the parabola and identify the vertex, the focus, the directrix, and the length of the latus

rectum.

24 2 13y x x

Vertex_____________ Focus______________

Directrix___________ LR________________


2. Graph the parabola and identify the vertex, the focus, the directrix, and the length of the latus

rectum.

216 4 20y x x

Vertex_____________ Focus______________

Directrix___________ LR________________

3. Graph the parabola and identify the following parts.

21 3 25

8 4 8y x x

Opens_________ Vertex____________

A.O.S._________ Value of a__________

Focus _________ Directrix___________


Writing the Equation of a Parabola given the Focus and the Directrix

1. Focus (3 , 8)

Directrix is y = 4

2. Focus is (7 , -2)

Directrix is y= 4

Write the equation of a Parabola given the Vertex and the Focus.

1. Vertex is (-1,2)

Focus is (-1,0)

2. Vertex is (5,2)

Focus is (3,2)


Math 3 Unit 3 Day 6 - Optimization

Steps:

1. Set up two equations (do not use x and y).

2. Solve the equation without the work max/min in it for any variable.

3. Substitute that expression into the other equation.

4. Know what your variables represent.

5. Enter into the calculator for y= and find the maximum or minimum.

6. Answer the question.

Examples:

1. Find two positive numbers whose sum is 36 and whose product is a maximum.

2. Find two numbers whose difference is 8 and whose product is a minimum.

3. A rectangle has a perimeter of 40 meters. Find the dimensions of the rectangle with the maximum

area.


4. Loni has 48 feet of fencing to make a rectangular dog pen. If a house is used for one side of the pen,

what would be the length and width for maximum area?

5. The circulation of the Charlotte Observer is 50,000. Due to increased production costs, the council

must increase the current price of 50 cents a copy. According to a recent survey, the circulation of the

newspaper will decrease 5000 for each 10 cent increase in price. What price per copy will maximize the

income from the newspaper?

6. A baseball stadium normally can sell 10,000 hotdogs at a game if they sell them for $4.00 each. The

also notice that if they raise the price $0.25 they will sell 500 fewer hotdogs. Determine the price they

should charge to maximize revenue.

7. Marsha is making a box to collect toys for the school toy drive. She cuts a 5 centimeter square from

each corner of a rectangular piece of cardboard and folds the sides up to make a box. If the perimeter

of the bottom of the box must be 50 centimeters, what should the length, width, and height of the box

be for a maximum volume?

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