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Algebra 2 Solve Quadratic Equations using Factoring - Notes

Name:_______________________________________ Date:____________________________________________________________________________________________________NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable

Vocabulary: Factor: breaking up the equation into binomials

Solution, zeros, x – intercept : solving the equation for x.

Quadratic Equation standard form is ax2 + bx + c (a, b, c are real numbers)

Part 1) Factor using GCF Method

1) 6 and 12 2) 6 and 15

3) 8x2 + 2x 4) 3x3y – 6x2y2 – 3xy3

Part 2 – All Trinomials – Factor using KrissKross, state the x-intercepts

1) x2 + 5x + 4 2) x2 – 8x + 12

3) 6x2 + 5xy – 4y2 4) 2x2 + 7xy – 4y2

5) x2 + 2x + 5 6) 6x4 + 11x2 – 10



Part 3 – Find the solution using square root What is different with these equations compared to part 2

equations?

Difference of Square (DOS)

u2 - v2 = (u + v)(u - v)

1) r2 – 9 2) x2 – 25

3) 100 – m2 4) 9x2 – 1

5) 49x2 – 100 6) 3d2 – 48

7) 11x2 – 10 8) 2x2 – 15

Part 4 – Factor by Quadratic formula What is different with these equations compared to part 2

equations?

Quadratic Formula: Discriminant

when given ax2 + bx + c = 0 a, b, c are the coefficients

Find the Discriminant (b2 – 4ac) to determine number and type of solution you will have.

Examples:

1: 3x2 + 8x - 14 = 0a) a = b = c = b) Disc. =

c) solution =

d) factors:

2: 4x2 + 20x + 25 = 0

a) a = b = c =

b) Disc. =

c) zeros =

d) factors:

3) x2 - 9x = 18 4) 25x2 + 9 = 30x



Recall….. Quadratic Formula: Discriminant

when given ax2 + bx + c = 0 a, b, c are the coefficients

Find the Discriminant (b2 – 4ac) to determine number and type of solution you will have.

Determine the type and number of solutions:

1: x2 + 6x + 8 = 0 2: x2 + 6x + 9 = 0 3: x2 + 6x = -10

Practice: 1) 2x2 + 4x - 4 = 0 2) 8x2 = 9x – 11 3) 3x – 5x2 + 1 = 6 – 7x


Name:_______________________________________ Date:____________________________________________________________________________________________________NJCCCS: N.CN.A.1Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.N.CN.A.2Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers

Complex numbersCan you think of a number to make this equation true?

Imaginary number: i =

i2 = -1

Examples:

Examples:

1) 2x2 + 11 = -37 2) –x2 + 4x = 5

Complex Number Standard Form: a + bi (real part + imaginary part)Operations with complex numbers (+, -, x):Combine real parts and combine imaginary parts.

1) (8 - i) + (5 + 4i) 2) (6 + 7i) + (1 - 3i)

3) (7 - 6i) - (3 - 6i) 4) (3 + 7i) - (8 - 2i)

5) 6i(4 – 2i)


Name:_______________________________________ Date:____________________________________________________________________________________________________

http://www.corestandards.org/Math/Content/HSN/CN/A/1/

http://www.corestandards.org/Math/Content/HSN/CN/A/2/

NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable

Part 6: Parabolas – Graphs of Quadratic Equations

y = ax2 + bx + c these quadratic equations form a shaped graph called a PARABOLA.

Ex A) y = x2

Step 1: Identify a, b, c a = b = c =

Step 2: If “a” is positive, then parabola opens up . Vertex is a minimum If “a” is negative, then parabola opens down . Vertex is a maximum

Step 3: Identify vertex (Xv, Yv) - ordered pair of turning point

Xv (x coordinate) =

To find Yv (y coordinate), substitute Xv into the equation.

V =

Sketch graph

X y

State x-intercepts:

1: y = x2 + x – 12a) a = b = c =

b) which way will the graph open? Min or max?

c) vertex:

d) sketch graph:

X y

State x-intercepts:

2: y = 2x2

a) a = b = c =


c) vertex:

d) sketch graph

X y

State x-intercepts:

3: -x2 - 9x + 8 = 0

a) a = b = c =


c) vertex:

d) sketch graph

X y

State x-intercepts:

4: 2x2 – 8x + 6 = 0

a) a = b = c =


c) vertex:

d) sketch graph

X y

State x-intercepts:

5: x2 – 2x - 1 = 0

a) a = b = c =


c) vertex:

d) sketch graph

X y

State x-intercepts:

6: 3x2 – 18x + 20 = 0

a) a = b = c =


c) vertex:

d) sketch graph

X y

State x-intercepts:



Word problems:

1) A ball is thrown, its height after t seconds is given by the function h(t) = 7 + 6t – t2 , where t is the time in seconds after the object is propelled and h is the object’s height in feet.

a) Draw a picture of the situation.

b) How many seconds after the ball is thrown will the object hit the ground?

c) What is the time for it to reach maximum height?

d) What is the maximum height?

e) What is the height at 2 seconds?

2) Given the area of a patio, sketch and label a picture with a possible expression for the width and length.

a) Area = 3x2 – 15x b) Area = 9 + 54x2

Is it possible to build the patio? Is it possible to build the patio? Yes No Yes NoExplain: Explain:



Part 6: Parabolas – Graphs of Quadratic Equations Graph using the calculator.

Ex A) y = x2

Step 1: load the equation into “y =” (remember to clear out all equations in there)


V( , ) is the vertex a min or max?What is the value of the min or max?

Step 3: fill in some values for table and state x- intercepts

x y

Step 4: sketch graph.

1: y = x2 + 1Step 1: load the equation




x y


2: y = -x2 + 2

Step 1: load the equation




x y

Step 4: sketch graph.3: y = x2 - 2x





x y


4: y = -2x2 – 4x + 7





x y

Step 4: sketch graph.5: y = -x2 + 3





x y


6: y = .33x2 – 2x + 4





x y



Name:_______________________________________ Date:_____________

_______________________________________________________________________________________NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable

Completing the Square

When solving a Quadratic Equation you need to factor it first. This is a short cut to factoring.

If you have a Perfect Square Trinomial (PST) then:x2 + 2x + 1 = (x + 1)2 = (x + 1)(x + 1)

How do you create a PST?Take half the 2nd term and then square it.

1) x2 + 12x + _____

2) x2 + 14x + _____

3) x2 - 20x + _____

Solve the Quadratic Equation.1) x2 + 12x + 36 = 0

2) x2 + 14x + 49 = 0

3) x2 - 20x + 100 = 0

Solve the Quadratic Equation by completing the square. Goal: Try to make a PST.

Step 1: Is left side x2 + bx? NO - move all numbers to right. Step 2: Add the number to make a PST to BOTH sides.Step 3: FactorStep 4: Take square root of BOTH sides (one = +, one = -)Step 5: solve

1) x2 + 2x = 0

2) x2 + 6x = 16

3) x2 - 2x – 8 = 0

4) x2 + 8x + 10 = 0

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