Algebra Honors ( Period 4)

Simplifying Radical Expressions 10-2

SQUARE ROOTS (RADICALS)

MAIN CONCEPT:

Square rooting "undoes" squaring!   It's the inverse operation!!!... Just as subtraction undoes addition     Just as division undoes multiplication

If you square a square root:
(√243)² = 243 (what you started with)

If you square root something squared:√243² = 243 (what you started with)
If you multiply a square root by the same square root:
(√243)(√243) = 243 (what you started with)

IN SUMMARY:

(√243)² = √243² = (√243)(√243) = 243

1) RADICAL sign: The root sign, which looks like a check mark.

If there is no little number on the radical, you assume it's the square root
But many times there will be a number there and then you are finding the root that the number says.

For example, if there is a 3 in the "check mark," you are finding the cubed root.
One more example: The square root of 64 is 8. The cubed root of 64 is 4. The 6th root of 64 is 2.

2)RADICAND : Whatever is under the RADICAL sign
In the example above, 64 was the radicand in every case.

3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power  to get the RADICAND (whatever is under the radical sign) In the example above, the roots were 8, 4, and 2.

4) SQUARE ROOTS: (What we primarily cover in Algebra I) The number that is squared to get to the radicand.

Every POSITIVE number has 2 square roots - one positive and one negative.
Example: The square root of 25 means what number squared = 25
Answer: Either positive 5 squared OR negative 5 squared = 25

5) PRINCIPAL SQUARE ROOT: The positive square root.

Generally, we assume the answer is the principal square root unless there is a negative sign in front of the radical sign or you’re solving to find both roots for a quadratic (We’ll deal with that in Chapter 13-2).

6) ± sign in front of the root denotes both the positive and negative roots at one time!

Example: √ 25 = ±5

7) ORDER OF OPERATIONS with RADICALS: Radicals function like parentheses when there is an operation under the radical. In other words, if there is addition under the radical, you must do that first (like you would do parentheses first) before finding the root.

EXAMPLE: √ (36 + 64) = 10 not 14!!!!

First add 36 + 64 = 100

Then find √100 = 10

Radicals by themselves function as exponents in order of operations
(that makes sense because they undo exponents).
Actually, roots are FRACTIONAL EXPONENTS!
Square roots = 1/2 power,
Cubed roots = 1/3 power,
Fourth roots = 1/4 power, etc.

So √25 = 25^½ = 5

EXAMPLE: 3 + 4√25
you would do powers first...in this case square root of 25 first!|
3 + 4(5)
Now do the multiplication
3 + 20
Now do the addition
23

8) THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!

EXAMPLE: √ 5² = 5

√ (a -7)² = a - 7

RATIONAL SQUARE ROOTS: Square roots of perfect squares are RATIONAL

REVIEW OF NUMBER SYSTEMS:

Rational numbers are decimals that either terminate or repeat

which means they can be restated into a RATIO a/b of two integers a and b where b is not zero.

Natural numbers: 1, 2, 3, ... are RATIOnal because you can put them over 1

Whole numbers: 0, 1, 2, 3,....are RATIOnal because you can put them over 1

Integers: ....-3, -2, -1, 0, 1, 2, 3,....are RATIOnal because you can put them over 1

Rational numbers = natural, whole, integers PLUS all the bits and pieces in between that can be expressed as repeating or terminating decimals: 2/3, .6, -3.2, -10.7 bar, etc.

Real numbers: all of these!

In Algebra II you will find out that there are Imaginary Numbers!
Square roots of NEGATIVE numbers are IMAGINARY

IRRATIONAL SQUARE ROOTS:  Square roots of a nonperfect squares are IRRATIONAL -

They cannot be stated as the ratio of two integers -
As decimals, they never terminate and never repeat -
you round them and use approximately sign.

MOST FAMOUS OF ALL IRRATIONAL NUMBERS IS PI!

  

note-- Square roots are MOSTLY IRRATIONAL!

There are fewer perfect squares than nonperfect!

Here are some perfect squares: 0, 4, 9, 16, 25, 36, etc.

PERFECT SQUARES CAN ALSO BE TERMINATING DECIMALS!

EXAMPLE: √( .04) is rational because it is ± .2

But all the square roots in between these perfect squares are IRRATIONAL

For example, the square root of 2, the square root of 3, the square root of 5, etc.
You can estimate irrational square roots.

For example, the square root of 50 is close to 7 because the square root of 49 is 7.
You can estimate that the square root of 50 is 7.1 and then square 7.1 to see what you get. 

If that's too much, try 7.05 and square that.
This works much better with a calculator! 

And obviously, a calculator will give you irrational square roots to whatever place your calculator goes to.
Remember: These will never end or repeat

(even though your calculator only shows a certain number of places physically!)

SIMPLIFYING RADICALS

SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:

A simplified radical expression is one where there is no perfect square left under the radical sign
You can factor the expression under the radical to find any perfect squares in the number:

EXAMPLE: √50 = √(25 * 2)

Next, simplify the SQRT of the perfect square and leave the nonperfect factor under the radical:

√(25 * 2) = √25 * √2 = 5√2

We usually read the answer as "5 rad 2"

HELPFUL HINTS:
When you are factoring the radicand,
you're looking for the LARGEST PERFECT SQUARE
that is a FACTOR of the radicand.
So start with:
Does 4 go into it?
Does 9 go into it?
Does 16 go into it?
Does 25 go into it?
etc.

Another method: Inverted Division or Factor Trees
Factor the radicand completely into its prime factors (remember this from Pre-Algebra?)
Find the prime factorization either way in order from least to greatest.
Circle factors in PAIRS
Every time you have a pair, you have a factor that is squared!
Then, you can take that factor out of the radical sign.
Remember that you are just taking one of those factors out!

Example: √ 250
Prime factorization = 2 x 5 x 5 x 5
Circle the first two 5's
5 x 5 is 25 and so you can take the square root of 25 = 5 out of the radicand
Everything else is not in a pair (squared) so it must remain under the radical

Final answer: √ 250 = 5√(2 x 5) = 5 √10

If there is a number in front of the radicand, simply multiply it by whatever you take out.

Algebra ( Period 5)

Transformations of Quadratic Functions  9-3

Transformations change the position or size of any figure.
In Math 8 we are doing this with geometric figures like triangles and parallelograms. In Algebra we will do this with parabolas.

TRANSLATION:
The parabola stays the same size but simply SLIDES to the left, right, up or down (or even a combination—like up and to the right)

VERTICAL TRANSLATION: A vertical translation up or down happens with you add a constant (k) to a parent function. For example the parent graph y = x² has the vertex at the origin but if you add 4 to the it,
 y = x² + 4à the vertex moves up 4 on the y axis
If you add -4  { y = x² + (-4)  or simply y = x² - 4}  it moves down 4 on the y axis.

HORIZONTAL TRANSLATION: A horizontal translation right or left happens when you add a constant (h) to the  x-value of the parent function.
We place that h value in a ( ) with the x variable:
(x –h)² would be a slide RIGHT

(x + h)² would be a slide LEFT

COMBINATION TRANSLATION: Together, both types of slides are shown by this formula ( known as the VERTEX FORMAT)  f(x) = (x –h)² +k
Notice that since the formula has –h, it moves in the opposite direction right or left. Since the formula is  +k   it moves in the same direction up or down!

DILATION: This type of transformation makes the graph either narrower or wider. As “a” is a smaller and smaller fraction/decimal the parabola gets wider and wider. AS “a” gets bigger—the parabola gets narrower.

REFLECTION: This transformation FLIPS the parabola upside down. This happens when “a” is negative.

So now the full VERTEX FORM of a parabola/quadratic is

y = a(x - h)² + k

Algebra Honors ( Period 4)

Transformations of Quadratic Functions  9-3

Transformations change the position or size of any figure.

In Math 8 we are doing this with geometric figures like triangles and parallelograms. In Algebra we will do this with parabolas.

TRANSLATION:
The parabola stays the same size but simply SLIDES to the left, right, up or down (or even a combination—like up and to the right)

VERTICAL TRANSLATION: A vertical translation up or down happens with you add a constant (k) to a parent function. For example the parent graph y = x² has the vertex at the origin but if you add 4 to the it,
y = x² + 4à the vertex moves up 4 on the y axis
If you add -4  { y = x² + (-4)  or simply y = x² - 4}  it moves down 4 on the y axis.

HORIZONTAL TRANSLATION: A horizontal translation right or left happens when you add a constant (h) to the  x-value of the parent function. We place that h value in a ( ) with the x variable:
(x –h)² would be a slide RIGHT

(x + h)² would be a slide LEFT

COMBINATION TRANSLATION: Together, both types of slides are shown by this formula ( known as the VERTEX FORMAT)  f(x) = (x –h)² +k

Notice that since the formula has –h, it moves in the opposite direction right or left. Since the formula is  +k   it moves in the same direction up or down!

DILATION: This type of transformation makes the graph either narrower or wider. As “a” is a smaller and smaller fraction/decimal the parabola gets wider and wider. AS “a” gets bigger—the parabola gets narrower.

REFLECTION: This transformation FLIPS the parabola upside down. This happens when “a” is negative.

So now the full VERTEX FORM of a parabola/quadratic is

y = a(x - h)² + k

Algebra ( Period 5)

Solving Quadratic Equations by Graphing 9-2

After putting the function in standard form ( ax² + bx + c) make the F(x) or y = 0 to find the x-intercept(s) of the parabola! We already know this!

x- intercepts = the roots = the zeros = the solutions of the quadratic.

One of three things will happen when you graph the parabola

1) it will go through the x axis twice (the vertex is below if it is a happy face or above it it’s  sad face.)  TWO REAL SOLUTIONS   TWO REAL ROOTS

2) It will have only one intercept (the VERTEX is on the x-axis) ONE REAL SOLUTION ONE REAL ROOT ( actually called a DOUBLE ROOT)

3) It will have NO x-intercepts ( the vertex is above if it is a happy face or below if it’s a sad face)  NO REAL SOLUTION   NO REAL ROOTS

If the x-intercept(s) are not an integer, we can estimate the roots OR use a calculator to find them exactly ( finding the zeros)

We sometimes just say that they are between the two integers that we find them on the graph!
We can also estimate to the tenths by making a table.

HOWEVER we will learn other methods in this chapter as we go along!

Algebra Honors ( Period 4)

Solving Quadratic Equations by Graphing 9-2

After putting the function in standard form ( ax² + bx + c) make the F(x) or y = 0 to find the x-intercept(s) of the parabola! We already know this!

x- intercepts = the roots = the zeros = the solutions of the quadratic.

One of three things will happen when you graph the parabola

1) It will go through the x axis twice (the vertex is below if it is a happy face or above it it’s  sad face.)  TWO REAL SOLUTIONS   TWO REAL ROOTS

2) It will have only one intercept (the VERTEX is on the x-axis) ONE REAL SOLUTION ONE REAL ROOT ( actually called a DOUBLE ROOT)

3) It will have NO x-intercepts ( the vertex is above if it is a happy face or below if it’s a sad face)  NO REAL SOLUTION   NO REAL ROOTS

If the x-intercept(s) are not an integer, we can estimate the roots OR use a calculator to find them exactly ( finding the zeros)

We sometimes just say that they are between the two integers that we find them on the graph!
We can also estimate to the tenths by making a table.

HOWEVER we will learn other methods in this chapter as we go along!

Algebra Honors ( Period 4)

Graphing Quadratic Functions 9-1

The standard form of a quadratic ( 2^nd degree function)  is

ax² + bx + c

The shape it makes when you graph it is called a parabola
Parabolas are symmetric around a line called the axis of symmetry. (AoS)

 The axis goes through a point on the parabola called the vertex

 The vertex is the maximum if the graph is a sad face and a minimum if it’s a happy face.
The parent graph is the simplest form of the graph.
For quadratics, the parent graph is  y = x²

 Note: that while b and c can both be ZERO, “a” can NEVER be zero for a quadratic.
WHY?      Then it would be a linear equation.

if a > 0 ( positive) the parabola opens upward ( happy face)
if a < 0 ( negative) the parabola opens downward ( sad face)

What happens as the “a” coefficient gets really big or really small ( fraction / decimal)?
The larger the “a” coefficientà the narrower the parabola.
The smaller (fractional/decimal) the “a” coefficientà the wider the parabola

{THINK: what happened when the “m” (or slope) coefficient got big? The slope got steeper. So, similarly, both sides of the U get steeper at the same time!!}

{NOW THINK: what happened when the “m” (or slope) coefficient got tiny ( or fractional)? The slope became what we called a bunny slope. So, similarly, both sides of the U get to be bunny slopes at the same time!!}

The x value of the vertex as well as the axis of symmetry are the SAME:

a and be are from the standard form of the quadratic.
Notice when b is missing ( b = 0), the vertex is always on the y-axis!! and therefore the axis of symmetry ( AoS) is also the y-axis.

 The c value is the y-intercept ( when b=0) because when x = 0 you are on the y axis.
f(x) = ax² + bx+ c
f(0) = a(0)² + b(0) + c
f(0) = c = the y value  … and in this case the y-intercept!

The domain of a quadratic (parabola) that is a function (opens up or down) is ALL REAL NUMBERS
The range depends on which way it opens:
If it opens up, the range will be   

      


If it opens down, the range will be    

You can graph quadratics exactly the same way you graphed lines—plug in your choice of an x value and use the equation to find your y value

 Because it is a U shape, you should graph at least 5 points as follows:

FIRST:  make sure the equation is in standard form!
y must be isolated on one side and then you can read the a and b coefficients

 y = ax² + bx + c

Point 1) the vertex: the minimum value of the smile or the maximum value of the frown
the x value of the VERTEX is  -b/2a
Plug that into the equation and find the y value of the vertex
Next draw the AXIS OF SYMMETRY (AoS)  

Notice- this is a linear equation! It is a line through the vertex that is parallel to the y-axis!

Point 2) Pick an x value immediately to the right or the left of the AXIS OF SYMMETRY (AoS)   and find its “y” by plugging into the equation.

Point 3)  Pick another x value one step farther on that same side of the AXIS OF SYMMETRY (AoS)  as point 2—and find its “y” by plugging into the equation.

Points 4 and 5) Graph the mirror image of both Point 2 and Point 3 on the other side of the AXIS OF SYMMETRY (AoS) by counting from the of the AXIS OF SYMMETRY (AoS)

You can choose to do two more points—if you want—but your points are joined in a SMOOTH U SHAPE ( not a V shape) and extend the lines with arrows on each end!

Example:   f(x) = -3x²  or just             y = -3x²

Notice right away—the “a” coefficient is negative so this is a frown face… a sad face! You know which way the graph will open! (Downward)

The x value of the vertex is –b/2a  a = -3 but b = 0 ( it is missing!)
so the x value of the vertex is –b/2a = -0/2(-3) = 0    Plug that back into the function f(0) = -3(0²) = 0

(Or just begin to realize that if b is missing the vertex is always on the y axis and the  AXIS OF SYMMETRY (AoS) will always be x = 0. )

In this case because there is NOT a “c” in the function, the vertex is ( 0,0) the origin.
The domain is all real numbers

The range is y ≤  0

To graph this function:

1) Graph the vertex ( 0,0)

2) Draw the AXIS OF SYMMETRY (AoS) a dotted line at x = 0  ( actually this is the y-axis so use a different colored pencil to indicate the AoS)

 3) Pick an x value immediately to the right of the AoS, x = 1 works so plug that into the equation to find its y value
y = -3(1²) = -3 PLOT ( 1, -3)

 4) Pick the next x value x = 2 and  plug that into the equation to find its y value
y = -3(2²) = -12 PLOT ( 2, -12)

 5) Count the same 1 step to the LEFT of the AoS as at the same y value of ( 1, -3) That would be ( -1, -3) and plot that point!

 6) Count the same 2 steps to the LEFT of the AoS at the same y value of ( 2 -12) That would be ( -2, -12) and plot that point

 7) Connect with a smooth U shape and extend with arrows!

Algebra (period 5)

Graphing Quadratic Functions 9-1

The standard form of a quadratic ( 2^nd degree function)  is

ax² + bx + c

The shape it makes when you graph it is called a parabola
Parabolas are symmetric around a line called the axis of symmetry. (AoS)

 The axis goes through a point on the parabola called the vertex

 The vertex is the maximum if the graph is a sad face and a minimum if it’s a happy face.
The parent graph is the simplest form of the graph.
For quadratics, the parent graph is  y = x²

 Note: that while b and c can both be ZERO, “a” can NEVER be zero for a quadratic.
WHY?      Then it would be a linear equation.

if a > 0 ( positive) the parabola opens upward ( happy face)
if a < 0 ( negative) the parabola opens downward ( sad face)

What happens as the “a” coefficient gets really big or really small ( fraction / decimal)?
The larger the “a” coefficientà the narrower the parabola.
The smaller (fractional/decimal) the “a” coefficientà the wider the parabola

{THINK: what happened when the “m” (or slope) coefficient got big? The slope got steeper. So, similarly, both sides of the U get steeper at the same time!!}

{NOW THINK: what happened when the “m” (or slope) coefficient got tiny ( or fractional)? The slope became what we called a bunny slope. So, similarly, both sides of the U get to be bunny slopes at the same time!!}

The x value of the vertex as well as the axis of symmetry are the SAME:

a and be are from the standard form of the quadratic.
Notice when b is missing ( b = 0), the vertex is always on the y-axis!! and therefore the axis of symmetry ( AoS) is also the y-axis.

 The c value is the y-intercept ( when b=0) because when x = 0 you are on the y axis.
f(x) = ax² + bx+ c
f(0) = a(0)² + b(0) + c
f(0) = c = the y value  … and in this case the y-intercept!

The domain of a quadratic (parabola) that is a function (opens up or down) is ALL REAL NUMBERS
The range depends on which way it opens:
If it opens up, the range will be   

      


If it opens down, the range will be    

You can graph quadratics exactly the same way you graphed lines—plug in your choice of an x value and use the equation to find your y value

 Because it is a U shape, you should graph at least 5 points as follows:

FIRST:  make sure the equation is in standard form!
y must be isolated on one side and then you can read the a and b coefficients

 y = ax² + bx + c

Point 1) the vertex: the minimum value of the smile or the maximum value of the frown
the x value of the VERTEX is  -b/2a
Plug that into the equation and find the y value of the vertex
Next draw the AXIS OF SYMMETRY (AoS)  

Notice- this is a linear equation! It is a line through the vertex that is parallel to the y-axis!

Point 2) Pick an x value immediately to the right or the left of the AXIS OF SYMMETRY (AoS)   and find its “y” by plugging into the equation.

Point 3)  Pick another x value one step farther on that same side of the AXIS OF SYMMETRY (AoS)  as point 2—and find its “y” by plugging into the equation.

Points 4 and 5) Graph the mirror image of both Point 2 and Point 3 on the other side of the AXIS OF SYMMETRY (AoS) by counting from the of the AXIS OF SYMMETRY (AoS)

You can choose to do two more points—if you want—but your points are joined in a SMOOTH U SHAPE ( not a V shape) and extend the lines with arrows on each end!

Example:   f(x) = -3x²  or just             y = -3x²

Notice right away—the “a” coefficient is negative so this is a frown face… a sad face! You know which way the graph will open! (Downward)

The x value of the vertex is –b/2a  a = -3 but b = 0 ( it is missing!)
so the x value of the vertex is –b/2a = -0/2(-3) = 0    Plug that back into the function f(0) = -3(0²) = 0

(Or just begin to realize that if b is missing the vertex is always on the y axis and the  AXIS OF SYMMETRY (AoS) will always be x = 0. )

In this case because there is NOT a “c” in the function, the vertex is ( 0,0) the origin.
The domain is all real numbers

The range is y ≤  0

To graph this function:

1) Graph the vertex ( 0,0)

2) Draw the AXIS OF SYMMETRY (AoS) a dotted line at x = 0  ( actually this is the y-axis so use a different colored pencil to indicate the AoS)

 3) Pick an x value immediately to the right of the AoS, x = 1 works so plug that into the equation to find its y value
y = -3(1²) = -3 PLOT ( 1, -3)

 4) Pick the next x value x = 2 and  plug that into the equation to find its y value
y = -3(2²) = -12 PLOT ( 2, -12)

 5) Count the same 1 step to the LEFT of the AoS as at the same y value of ( 1, -3) That would be ( -1, -3) and plot that point!

 6) Count the same 2 steps to the LEFT of the AoS at the same y value of ( 2 -12) That would be ( -2, -12) and plot that point

 7) Connect with a smooth U shape and extend with arrows!

Algebra Honors ( Period 4)

Factoring Trinomials 8-6

We  will factor them and then put our fully factored form on the same graph as the simplified form—What do you think will happen if we factored the trinomial (quadratic, 2^nd degree polynomial) correctly?

Factoring Trinominals with a POSITIVE sign as the SECOND SIGN

You are trying to turn a trinomial back to the two binomials that were multiplied together to get it!

Always check your factoring by FOILing back!

There is a simple method for foiling basic trinomials

1) set up your two sets of  {{HUGS}  (    ) (    )
2) When the last sign is positive then BOTH SIGNS in each of the {{HUGS}} are the SAME—they could be Both POSITIVE or they could be BOTH NEGATIVE!!
3) How do you know what the 2 signs are? It is whatever the SIGN is of the MIDDLE ( SECOND) term!! PUT THAT SIGN in both {{HUGS}} 
4) TO UNFOIL ( Factor) you will need to find 2 factors that MULTIPLY to the Last Term & ALSO ADD to the Middle Term
To help you do this I suggest you use an X Put the product in the top of the X and the sum in the bottom of the X . Then figure out the correct factors on the left and right of the X. It gets easier with practice! I promise

Understand that this is really just an educated guess and check!

EXAMPLE 1
x² + 8x + 15

(   )(   )

Two set of {{HUGS}} are set up
THINK: Last sign is + so both of the signs inside the parentheses are the same
THINK: Middle sign is + so both of the signs are  +
( + )( + )

You know that the “F” in FOIL means that both first terms must be x so go ahead an place the x in each
(x + )(x + )
You need to get the “L” in FOIL You need 2 factors whose product is 15 Like 1 and 15 or 2 and 5
If you use the “X” from above  YOU can see it!

But if you are having difficulty realize you need to add to the I and O in FOIL which means that the two factors must add to 8

___ ∙ ____ = 15

____ + ___ = 8

Since 3 + 5 = 8 this must be the two factors that work

3 ∙ 5= 15
3 + 5 = 8

(x + 5)(x + 3)

In this case it doesn’t matter which factor you put in the first set of  {{HUGS}} because they are the same sign. (I always tend to put the larger number in the first parentheses for a reason that you will see tomorrow.)  Now FOIL to see if we are right!!

EXAMPLE 2
x² - 8x + 15
Looks the same with one difference—the middle term is negative

(   )(   )

Two set of {{HUGS}} are set up

THINK: Last sign is + so both of the signs inside the parentheses are the same
THINK: Middle sign is - so both of the signs are  -
( - )( - )

You need to get the “L” in FOIL You need 2 factors whose product is 15 Like 1 and 15 or 2 and 5
If you use the “X” from above  YOU can see it!
But if you are having difficulty realize you need to add to the I and O in FOIL which means that the two factors must add to 8

-___ ∙ -____ = 15

-____ + -___ = -8

Since -3 + -5 = -8 this must be the two factors that work

-3 ∙ -5= 15
-3 + -5 = -8

(x - 5)(x - 3)

I actually just ignore the signs which making an educated guess because I have already put the negative signs in both parentheses so I have taken care of the negatives. It is up to you which way you are most comfortable… But once you make yup your mind--> stick with that method

EXAMPLE 3
Same problem but now with a “y” on the middle and last terms
Looks the same with one difference—the middle term is negative  Simply use the same factorization as above and add the y

x² - 8xy + 15y²

Simply use the same factorization as above and add the y

(x- 5y )(x – 3y)
Always FOIL back to check!
Remember the Little nose and the Big Smile—to check the middle term—that is usually were students make a mistake. It is the O and I in FOIL

Factoring Trinomials with a Negative Sign as the Second Term
This is a bit more complicated

1) set up your two sets of  {{HUGS}}  (    ) (    )
2) Look at that last sign, If it is NEGATIVE, then the signs in the {{HUGS}} must be DIFFERENT. Why? Because in order to get a negative when you multiply integers—one of them needs to be negative and the other must be positive. Remember that the last term is the product of the two last terms in FOILing.
3) Now look at the sign of the second term. It tells you “Who wins” meaning which sign must have the bigger number or which has the greater absolute value. Remember that the middle term is the SUM of the O and the I terms when FOILING.  Because these two terms have different signs, when you add them—you actually take the difference (subtract) and take the “bigger number’s”  sign.
PUT THAT SIGN IN THE FIRST PARENTHESES and always put the “bigger number” in the first set of {{HUGS}}
4) TO UNFOIL ( Factor) you will need to find 2 factors that MULTIPLY to the Last Term & ALSO SUBTRACT to the Middle Term  (YOU CAN STILL SAY YOU ARE ADDING BUT SINCE THEY ARE DIFFERENT SIGNS YOU ARE TAKING THE DIFFERENCE)

To help you do this I suggest you use an X Put the PRODUCT in the top of the X and the DIFFERENCE in the bottom of the X . Then figure out the correct factors on the left and right of the X. It gets easier with practice! I promise

Understand that this is really just an educated guess and check!

EXAMPLE 4
x²+ 2x – 15

(   )(   )

Two set of {{HUGS}} are set up
THINK: Last sign is - so both of the signs inside the parentheses are DIFFERENT
THINK: Middle sign is + so the POSITIVE WINS
( + )( - )

You know the “F” in FOIL means that both first terms must be x

(x + )(x - )

Now to get the “L” in FOIL you need 2 factors whose product is NEGATIVE 15

-1 and 15, 1 and -15 or -3 and 5 or 3 and -5

But, since the POSITIVE must win, according to the POSITIVE TERM of 2x, you  know that the bigger factor must be POSITIVE ( so it can win)
so either + 15 and -1 or +5 and -3 are the only choices
But, you also need to add to the I and the O in FOIL so pick the two factors that also ADD to POSITIVE 2
Therefore it has to be +5 and -3. 

+___ ∙ -____ = 15

+___ + -___ = 2

+5 ∙ -3 = 15
+5 + -3 = 2

so    (x +5 )(x - 3)

EXAMPLE 5
x²-  2x – 15
That’s the same as before EXCEPT the middle sign is now negative

(   )(   )

Two set of {{HUGS}} are set up

THINK: Last sign is - so both of the signs inside the parentheses are DIFFERENT
THINK: Middle sign is - so the NEGATIVE WINS
Its exactly the same but this time you need to get to a difference of -2
(x -5 )(x +3)

EXAMPLE 6
Same as the last example but this time with a “y”

x²-  2xy – 15y²

Same problem except that there are two variables—make sure to add the Y at the end—or set it up right from the start…(   )(   )

Two set of {{HUGS}} are set up

(x y)(x y)

Use the same technique

(x - 5y)(x + 3y)

Math 8 ( Period 1)

Use the Pythagorean Theorem 5-6

We use the Pythagorean Theorem to solve many real word problems:

The Pythagorean Theorem formula: a² + b² = c²

For example, the height of a ladder leaning on a building (that’s a right triangle)
The height of a tree if you know the height of the ladder leaning against it and how far the ladder is from the base of the tree

The length of a wire needed to support a flag pole if you know its height.

The height of a kite if you know the length of its string and how far it is away from where you are standing.

As long as you can visualize the problem as a RIGHT TRIANGLE you can use the formula to solve for an unknown measurement.

Example 1: How high a 10 foot ladder reaches on a house
A 10 ft ladder is placed on a house  5 ft away from the base of the house.
Find how high up the house the ladder reaches
The ladder makes a right triangle with the ground being one leg, the house being the other and the ladder is actually the hypotenuse.
You need to find the distance on the house so you are finding one of th legs in the right triangle.

Example 2: You are flying your kite for the kite project and you want to know how long the kite string must be so that it can reach a height of 13 ft in the air if you area standing 9 feet away from where the kite is in the air.  The string represents the hypotenuse. You know one leg is the height in the air ( 13 ft) and the other leg is how far on the ground you are standing away from where the kite  is flying ( 9ft) This time you need to find the hypotenuse

The Pythagorean Theorem formula: a² + b² = c²

Algebra ( Period 5)

Factoring Trinomial Squares 8-9

Chapter 8-4 Déjà vu  Foil the following:

(a +3)² ( which is called a binomial squared)
(a + 3)(a +3) = a² + 3a + 3a + 9 = a² + 6a + 9 ( called a trinomial square)

Again, you see that the middle term is DOUBLE the product of the two terms in the binomial
and the first and last terms are simply the squares of each term in the binomial

How to recognize that it is a Binomial Squared:

1) Is it a trinomial? ( if it’s a binomial it cannot be a binomial squared)
2) Are the first  and last terms POSITIVE?
3) Are the first and last terms PERFECTSQUARES?4) Is the middle term DOUBLE the product of the SQRTS of the 1^st and last terms?

If YES to ALL of these questions you have a trinomial square

To FACTOR a Trinomial Squarea² + 6a + 9 (called a trinomial square)

1) Put ONE set of {{HUGS} with an exponent of 2  (  )²
2) Put the sign of the middle term inside  (in this case its +)  (  +  )²
3) Find the SQRT’s of the first term & last terms; place them in parentheses (a +3)²4) Check by FOILing back

If the sign is negative in the middle, simple use a negative sign when you factor.