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STANDARDS OF LEARNING
CONTENT REVIEW NOTES
ALGEBRA I
3rd Nine Weeks, 2018-2019
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OVERVIEW
Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
for students and parents. Each nine weeksโ Standards of Learning (SOLs) have been identified and a detailed
explanation of the specific SOL is provided. Specific notes have also been included in this document to assist
students in understanding the concepts. Sample problems allow the students to see step-by-step models for
solving various types of problems. A โ โ section has also been developed to provide students with the
opportunity to solve similar problems and check their answers. The answers to the โ โ problems are
found at the end of the document.
The document is a compilation of information found in the Virginia Department of Education (VDOE)
Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE
information, Prentice Hall textbook series and resources have been used. Finally, information from various
websites is included. The websites are listed with the information as it appears in the document.
Supplemental online information can be accessed by scanning QR codes throughout the document. These will
take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the
document to allow students to check their readiness for the nine-weeks test.
The Algebra I Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of
questions per reporting category, and the corresponding SOLs.
Algebra I Blueprint Summary Table
Reporting Categories No. of Items SOL
Expressions & Operations 12 A.1
A.2a โ c
A.3
Equations & Inequalities 18 A.4a โ f
A.5a โ d
A.6a โ b
Functions & Statistics 20 A.7a โ f
A.8
A.9
A.10
A.11
Total Number of Operational Items 50
Field-Test Items* 10
Total Number of Items 60
* These field-test items will not be used to compute the studentsโ scores on the test.
It is the Mathematics Instructorsโ desire that students and parents will use this document as a tool toward the
studentsโ success on the end-of-year assessment.
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Laws of Exponents & Polynomial Operations A.2 The student will perform operations on polynomials, including
a) applying the laws of exponents to perform operations on expressions;
Monomial is a single term. It could refer to a number, a variable, or a product of a number and one or more variables. Some examples of monomials include:
14๐๐ยฒ โ 6๐ 1 1
2๐ฅยฒ๐ฆ๐งยฒ
When you multiply monomials that have a common base, you add the exponents.
Example 1: Multiply ๐2 โ ๐5
๐2 โ ๐5 = ๐2+5 = ๐7
This works because when you raise a number or variable to a power, it is like multiplying it by itself that many times. When you then multiply this by another power of the same number or variable, you are just multiplying it by itself that many more times.
Example 2: ๐ ๐๐ค๐๐๐ก๐ 43 ๐๐๐ 42๐๐ ๐๐ข๐๐ก๐๐๐๐๐๐๐ก๐๐๐ ๐๐๐๐๐๐๐๐ . ๐๐ ๐ ๐กโ๐๐ ๐ก๐ ๐ ๐๐๐๐๐๐๐ฆ 43 โ 42.
43 = 4 โ 4 โ 4 42 = 4 โ 4
43 โ 42 = 4 โ 4 โ 4 โ 4 โ 4 = 45
Example 3: Simplify 1
2๐ฅ๐ฆ3๐ง5 โ 14๐ฅ3๐ง โ ๐ฆ6๐ง2
(1
2 โ 14) (๐ฅ โ ๐ฅ3)(๐ฆ3 โ ๐ฆ6)(๐ง5 โ ๐ง โ ๐ง2)
7๐ฅ4๐ฆ9๐ง8 When you raise a power to a power, you multiply the exponents.
(32)4
32 โ 32 โ 32 โ 32
32+2+2+2
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Example 4: Simplify (๐2๐)4
(๐2๐)4 = ๐2โ4๐1โ4
๐8๐4
This means 3ยฒ times itself 4 times!
Scan this QR code to go to a video tutorial on
multiplying monomials!
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Example 5: Simplify (5๐5๐7๐2)3
53๐5โ3๐7โ3๐2โ3
125๐15๐21๐6
Often, you will be asked to multiply monomials and raise powers to a power. Make sure that you follow the ORDER OF OPERATIONS! Raise to powers first, then multiply.
Example 6: Simplify (โ2๐ฅยณ๐ฆ๐งยฒ)ยณ(2๐ฅยณ๐ฆ๐งยฒ)โด
(โ8๐ฅ9๐ฆ3๐ง6) โ (16๐ฅ12๐ฆ4๐ง8)
โ128๐ฅ21๐ฆ7๐ง14
Laws of Exponents Simplify each expression
1. 6๐3๐5(โ๐๐2)
2. (โ5๐ฅ2๐ฆ๐ง3)2
3. [(๐๐6๐3)2]4
4. (๐4๐2๐7)4 (13๐2๐)3 When you divide monomials with like bases, you will subtract the exponents.
Anything raised to the zero power is equal to ONE!
๐ฅ0 = 1 150 = 1 (โ235๐ฆ7)0 = 1 To find the power of a quotient, raise both the numerator and the denominator to the power. (Remember to follow the order of operations!)
(๐
๐)
5
= ๐5
๐5
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Example 7: ๐5๐7
๐4๐
๐5๐7
๐4๐= ๐5โ4 ๐7โ1 = ๐ ๐6
Example 8: (๐2๐๐5
๐2๐2 )3
(๐2๐๐5
๐2๐2)
3
= (๐2โ2๐๐5โ2)3 = (๐0๐๐3)3 = 13๐3๐9 = ๐3๐9
You will also see negative exponents in monomials. When you have a negative exponent, you will reciprocate that variable (move it to the other side of the fraction bar) and the exponent will become positive.
As an example: 2โ3 = .125 = 1
8 ๐๐ 2โ3 =
1
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When simplifying monomials with negative exponents, you can start by โflipping overโ all of the negative exponents to make them positive. Then, simplify.
Example 9: ๐โ3๐2๐โ5
๐โ7๐๐10
๐โ3๐2๐โ5
๐โ7๐๐10
๐7๐2
๐3๐๐10๐5
๐4๐
๐15
Example 10: (2๐ฅ2๐ฆโ4
3๐ฅ๐ฆ5 )โ3
(2๐ฅ2๐ฆโ4
3๐ฅ๐ฆ5)
โ3
= (3๐ฅ๐ฆ5
2๐ฅ2๐ฆโ4)
3
= (3๐ฅ๐ฆ5๐ฆ4
2๐ฅ2)
3
= (3๐ฆ9
2๐ฅ)
3
= 27๐ฆ27
8๐ฅ3
Remember that anything to the zero power equals 1!
Scan this QR code to go to a video tutorial on
dividing monomials!
Scan this QR code to go to a video tutorial on
simplifying monomials
with negative exponents!
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Exponents Laws of Exponents
Simplify each expression
5. 33๐5๐9
11๐6๐2
6. (๐ฅ4๐ฆ2
2๐ฅ๐ฆ)
3
7. (2๐ฅ๐ฆโ3)5
8. ๐4๐2๐โ5
๐6๐โ2๐0
9. 15๐๐โ2๐5
9๐โ4๐2๐2
10. (6๐ฅ4๐ฆ2๐งโ3
9๐ฅโ2๐ฆ0๐งโ1 )
โ2
Polynomials A.2 The student will perform operations on polynomials, including
b) adding, subtracting, and multiplying polynomials.
Adding and subtracting polynomials is the same as COMBINING LIKE TERMS. In order for two terms to be like terms, they must have the same variables and the same exponents.
Like Terms NOT Like Terms
5๐๐2, โ3๐๐2,2
3๐๐2 5๐๐2, โ3๐2๐,
2
3๐๐
Each of these terms contain an โ๐๐2 โ, therefore they are like terms.
Although these terms have the same variables, corresponding variables do not
have the same exponents. Therefore, these are NOT like terms.
Example 1: (2๐ฅ2๐ฆ + 5๐ฅ๐ฆ โ 7๐ฆ2) + (4๐ฅ2๐ฆ โ 10๐ฅ๐ฆ + 3๐ฆ2)
(2๐ฅ2๐ฆ + 4๐ฅ2๐ฆ) + (5๐ฅ๐ฆ โ 10๐ฅ๐ฆ) + (โ7๐ฆ2 + 3๐ฆ2)
6๐ฅ2๐ฆ โ 5๐ฅ๐ฆ โ 4๐ฆ2
Like terms are underlined here. Remember that each term takes the sign in front of it!
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Remember that if you are subtracting a polynomial, you are subtracting all of the terms (Therefore, you must distribute the negative to each term first!)
Example 2: (โ3๐๐ โ 5๐4๐2 + ๐) โ (4๐๐ โ 6๐)
โ3๐๐ โ 5๐4๐2 + ๐ โ 4๐๐ + 6๐
โ7๐๐ โ 5๐4๐2 + 7๐
Polynomials Simplify each expression
1. (๐๐๐2 โ 7๐๐2 + 12๐๐) + (4๐๐๐2 โ 3๐๐)
2. (4๐ + 9๐ โ 3๐ + 2๐) + (2๐ โ ๐ โ 5๐ + 3๐2)
3. (12๐ฅ2 โ 6๐ฅ๐ฆ + 9๐ฆ2) โ (3๐ฅ2 + ๐ฅ๐ฆ โ 4๐ฆ2)
4. (32๐๐2 + 5๐2๐ โ 21๐2) โ (๐๐ + 14๐2 โ 5๐2๐) 5. (2๐ก๐ข โ 8๐ข + 7๐ก) + (โ๐ก๐ข โ 4๐ก) โ (3๐ข + ๐ก โ 5๐ก๐ข) To multiply a polynomial by a monomial, simply distribute the monomial to each term in the polynomial. You will use the rules of exponents to simplify each term.
Example 3: 5๐ฅ (3๐ฅ2 โ 6๐ฅ๐ฆ + 2๐ฆ2)
(5๐ฅ โ 3๐ฅ2) โ (5๐ฅ โ 6๐ฅ๐ฆ) + (5๐ฅ โ 2๐ฆ2)
15๐ฅ3 โ 30๐ฅ2๐ฆ + 10๐ฅ๐ฆ2
Distribute the negative to everything in the second set of parentheses!
Then, COMBINE LIKE TERMS!
Scan this QR code to go to a video tutorial on
adding and subtracting polynomials.
Distribute the 5๐ฅ to each term.
Then, simplify each term
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Example 4: โ2๐2๐ (5๐๐3 โ 6๐2๐5 + ๐2๐ โ ๐๐3 ) (โ2๐2๐ โ 5๐๐3) + (โ2๐2๐ โ โ6๐2๐5 ) + (โ2๐2๐ โ ๐2๐ ) + (โ2๐2๐ โ โ๐๐3)
โ10๐3๐4 + 12๐4๐6 โ 2๐4๐2 + 2๐3๐4
โ8๐3๐4 + 12๐4๐6 โ 2๐4๐2 To multiply two polynomials together, distribute each term in the first polynomial to each term in the second polynomial. When you are multiplying two binomials together this may be called FOIL. FOIL stands for:
F โ First โ multiply the first term in each binomial together O โ Outer โ multiply the outermost term in each binomial together I โ Inner โ multiply the innermost term in each binomial together L โ Last โ multiply the last term in each binomial together (This is the exact same as distributing the first term, then distributing the second term) Donโt forget to combine like terms when possible. Example 5: (2๐ฅ + 5)(3๐ฅ โ 2) First Outer Inner Last
(2๐ฅ โ 3๐ฅ) + (2๐ฅ โ โ2) + (5 โ 3๐ฅ) + (5 โ โ2)
6๐ฅ2 โ 4๐ฅ + 15๐ฅ โ 10
6๐ฅ2 + 11๐ฅ โ 10
Example 6: (๐2 + 2๐2)(4๐ โ 3๐๐ + 6๐)
(๐2)(4๐ โ 3๐๐ + 6๐) + (2๐2)(4๐ โ 3๐๐ + 6๐)
4๐3 โ 3๐3๐ + 6๐2๐ + 8๐๐2 โ 6๐๐3 + 12๐3
Donโt forget to check for like terms!
Scan this QR code to go to a video tutorial on
multiplying monomials and polynomials.
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Example 7: (4๐ฆ โ 3)2 (4๐ฆ โ 3)(4๐ฆ โ 3) (4๐ฆ โ 4๐ฆ) + (4๐ฆ โ โ3) + (โ3 โ 4๐ฆ) + (โ3 โ โ3)
16๐ฆ2 โ 12๐ฆ โ 12๐ฆ + 9
16๐ฆ2 โ 24๐ฆ + 9
Polynomials
6. โ2๐๐2(4๐2๐ โ 3๐๐) 7. 2๐๐2๐ (4๐ + ๐ โ 3๐) + 5๐๐2๐ (6๐ โ 3๐ โ 5๐) 8. (6๐ฅ + 5)(6๐ฅ โ 5)
9. (5๐2๐ โ 2๐2)(4๐๐ + ๐)
10. (3๐ฅ๐ฆ โ 5๐ฅ)2 Factoring A.2 The student will perform operations on polynomials, including
c) factoring completely first- and second-degree binomials and trinomials in one variable.
The prime factorization of a number or monomial is that number or monomial broken
down into the product of its prime factors.
Example 1: Write the prime factorization of 18๐ฅ3
18๐ฅ3
18๐ฅ3๐ฆ2 = 2 โ 3 โ 3 โ ๐ฅ โ ๐ฅ โ ๐ฅ or 2 โ 32 โ ๐ฅ โ ๐ฅ โ ๐ฅ
To find the greatest common factor (GCF) of two or more monomials, break each down
into its prime factorization. The GCF is the product of all of the shared factors.
Remember that to square something means to multiply it by itself!
9 2
3 3
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Example 2: What is the greatest common factor of 9๐, 15๐, ๐๐๐ 6๐2
๐บ๐ถ๐น = 3๐
You can use the GCF to help you rewrite (factor) polynomials. If all of the terms in the
polynomial have common factors you can pull these factors out from the terms to factor
the polynomial.
Example 3: Factor 8๐ฅ2 + 20๐ฅ
8๐ฅ2 = 2 โ 2 โ 2 โ ๐ฅ โ ๐ฅ
20๐ฅ = 2 โ 2 โ 5 โ ๐ฅ
4๐ฅ (2๐ฅ + 5)
Example 4: Factor 15๐3 โ 15๐2
15๐3 = 3 โ 5 โ ๐ โ ๐ โ ๐ โ
โ30๐2 = โ1 โ 2 โ 3 โ 5 โ ๐ โ ๐ โ
15๐2 = 3 โ 5 โ ๐ โ ๐ โ
15๐2 (๐ โ 1)
Factoring
1. Write the prime factorization of 180๐2
2. Find the greatest common factor of 15๐ฅ2 ๐๐๐ 42๐ฅ
3. Factor 8๐3 + 14๐ โ 12
Circle each factor that they ALL have in common!
GCF = 2 โ 2 โ ๐ฅ = 4๐ฅ
Pull the GCF out from each term and rewrite. Check your work by distributing.
GCF = 3 โ 5 โ ๐ โ ๐ โ ๐ = 15๐2
Pull the GCF out from each term and rewrite. Check your work by distributing.
Scan this QR code to go to a video tutorial on
greatest common factors.
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Simplifying Radicals
A.3 The student will simplify a) square roots of whole numbers and monomial algebraic expressions;
To simplify a radical, you will pull out any perfect square factors (i.e. 4, 9, 16, 25, etc.)
โ18 = โ9 โ 2
The square root of 9 is equal to 3, so you can pull the square root of 9 from underneath
the radical sign to find the simplified answer 3 โ2 , which means 3 times the square
root of 2. You can check this simplification in your calculator by verifying that
โ18 = 3โ2 .
Another way to simplify radicals, if you donโt know the factors of a number, is to create
a factor tree and break the number down to its prime factors. When you have broken
the number down to all of its prime factors you can pull out pairs of factors for square
roots, which will multiply together to make perfect squares.
Example 5: Simplify โ128 โ2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2
2 โ 2 โ 2 โ2 = 8โ2
Example 6: Simplify 3โ32๐ฅ3๐ฆ
To simplify a root of a higher index, you pull out factors that occur the same number of
times as the index of the radical. As an example, if you are simplifying โ64๐75 , you
would only pull out factors that occurred 5 times, since 5 is the index of the root.
2 64
8 8
4 2 2 4
2 2 2 2
16 2 x x x y
4 4
2 2 2 2
3โ2 โ 2 โ 2 โ 2 โ 2 โ ๐ฅ โ ๐ฅ โ ๐ฅ โ ๐ฆ
3 โ 2 โ 2 โ ๐ฅโ2๐ฅ๐ฆ = 12๐ฅโ2๐ฅ๐ฆ
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Example 7: Simplify โ64๐73
โ2 โ 2 โ 2 โ 2 โ 2 โ 2 โ ๐ โ ๐ โ ๐ โ ๐ โ ๐ โ ๐ โ ๐3
2 โ 2 โ ๐ โ ๐ โ๐3
4๐2 โ๐3
Simplifying Radicals Simplify the following radicals.
4. โ4๐ฅ4๐ฆ3
5. 6๐โ15๐๐4๐3
6. โ48๐4๐23
Factoring Special Cases A.2 The student will perform operations on polynomials, including
c) factoring completely first- and second-degree binomials and trinomials in one variable.
A.3 The student will simplify
a) square roots of whole numbers and monomial algebraic expression b) cube roots of integers
Factoring Trinomials
To factor a trinomial of the form ๐ฅ2 + ๐๐ฅ + ๐, first find two integers whose sum is equal
to ๐, and whose product is equal to ๐ โ ๐ . You can start by listing all of the factors of ๐ โ ๐, and then see which two factors add up to the coefficient of ๐. Once you have determined which factors to use, you can put all of your terms โin a boxโ and factor the rows and columns.
Scan this QR code to go to a video tutorial on
simplifying radicals.
Because this is a cube root, I pulled out things that occurred 3 times.
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Example 1: Factor ๐ฅ2 + 6๐ฅ + 8 ๐ โ ๐ = 1 โ 8 = 8 So, we are looking for factors of 8 that add up to 6! Factors of 8 Sum of factors 1, 8 9 2, 4 6 Put terms โin a boxโ
Sometimes you will not be able to find factors of ๐ โ ๐ that sum to b. When this happens, the trinomial is PRIME.
Example 2: Factor 2๐ฅ2 + 5๐ฅ โ 2 ๐ โ ๐ = 2 โ โ2 = โ4 So, we are looking for factors of -4 that add up to 5! Factors of 8 Sum of factors 1, -4 -3 -1, 4 3 -2, 2 0 Nothing works, therefore this trinomial is PRIME When factoring, anytime the ๐ term is negative and the ๐ term is positive, your answer will have two minus signs!
First Term
(๐๐ฅ2)
One Factor (__๐ฅ)
Other Factor (__๐ฅ)
Last Term
(c)
๐ฅ2 2๐ฅ
4๐ฅ 8
Find the greatest common factor in each row and each column. These will give you your two binomials!
๐ฅ 4
๐ฅ 2
(๐ฅ + 4) (๐ฅ + 2)
Check your answer by FOIL-ing!
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Example 3: Factor 8๐ฅ2 โ 21๐ฅ + 10 ๐ โ ๐ = 8 โ 10 = 80 So, we are looking for factors of 80 that add up to โ21! Factors of 80 Sum of factors -4, -20 -24 -5, -16 -21
Example 4: Factor 3๐ฅ2 + 24๐ฅ + 45
Pull out a GCF first!! 3(๐ฅ2 + 8๐ฅ + 15) ๐ โ ๐ = 1 โ 15 = 15 So, we are looking for factors of 15 that add up to 8!
Factoring Special Cases Factor each of the trinomials below
1. ๐ฅ2 + 7๐ฅ + 12
2. 2๐ฅ2 โ 14๐ฅ๐ฆ โ 36๐ฆ2
3. 6๐ฅ2 + 17๐ฅ + 5
4. ๐ฅ2 โ 9๐ฅ + 1
8๐ฅ2 โ5๐ฅ
โ16๐ฅ 10
๐ฅ2 3๐ฅ
5๐ฅ 15
Find the greatest common factor in each row and each column. These will give you your two binomials!
๐ฅ
โ2
8๐ฅ โ5
(8๐ฅ โ 5) (๐ฅ โ 2)
Check your answer by FOIL-ing!
Scan this QR code to go to a video tutorial on
factoring trinomials.
Find the greatest common factor in each row and each column. These will give you your two binomials!
๐ฅ
5
๐ฅ 3
3(๐ฅ + 5) (๐ฅ + 3)
Check your answer by FOIL-ing! Donโt forget your GCF in the front.
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To solve a quadratic equation (i.e. find its solutions, roots, or zeros), set one side equal
to zero (put the quadratic in standard form), then factor. Set each factor equal to zero
to find the values for ๐ฅ that are the solutions to the quadratic.
Example 5: Find the zeros of ๐ฅ2 โ 18 = 7๐ฅ
Start by getting one side equal to zero and write in standard form.
๐ฅ2 โ 18 = 7๐ฅ
โ7๐ฅ โ 7๐ฅ
๐ฅ2 โ 7๐ฅ โ 18 = 0 Now factor the trinomial.
We are looking for factors of โ18 that add up to โ7. โ9 and 2 work!
(x + 2)(x โ 9) = 0 Set both factors equal to zero!
๐ฅ + 2 = 0 ๐๐๐ ๐ฅ โ 9 = 0
๐ฅ = โ2 ๐๐๐ 9 or {โ2, 9}
You can check your answer in your calculator by graphing the quadratic. The solutions
are the x-intercepts, so this graph should cross the x-axis at -2 and 9.
Factoring Special Cases Find the solution to each trinomial
5. ๐ฅ2 + 9๐ฅ + 20 = 0
6. 2๐ฅ2 + 6 = 7๐ฅ
7. 2๐ฅ3 + 10๐ฅ2 โ 10๐ฅ = 2๐ฅ
๐ฅ2 โ9๐ฅ
2๐ฅ โ18
๐ฅ
2
๐ฅ โ9
-2 9
Scan this QR code to go to a video tutorial on solving
trinomials by factoring.
17
Special Cases
A perfect square trinomial can be factored to two binomials that are the same, so you
can write it as the binomial squared.
๐2 + 2๐๐ + ๐2 = (๐ + ๐)2 ๐2 โ 2๐๐ + ๐2 = (๐ โ ๐)2
Example 6: Factor 4๐ฅ2 โ 24๐ฅ + 36
If your first and last terms are perfect squares you can check for a perfect square
trinomial. Take the square root of the first and last number and see if the product
of those is equal to ยฝ of the middle number.
โ4 = 2 ๐๐๐ โ36 = 6 6 โ 2 = 12 , ๐คโ๐๐โ ๐๐ 1
2 ๐๐ 24
Now that we know this case works, you can write the binomial factor squared
(2๐ฅ โ 6)2
Remember to check your answer by FOIL-ing the binomials back out!
Another special case is if the quadratic is represented as the difference of two perfect
squares (i.e. 4๐ฅ2 โ 16). If both the first and last term are perfect squares, and the two
terms are being subtracted their factorization can be written as (๐ + ๐)(๐ โ ๐). As an
example 4๐ฅ2 โ 16 = (2๐ฅ + 4)(2๐ฅ โ 4). Remember that you can check your work by
FOIL-ing.
Example 7: Factor completely 3๐ฅ2 โ 27
To begin, you should factor out a GCF. In this case it would be 3.
3(๐ฅ2 โ 9) Now you are left with a difference of squares!
3(๐ฅ + 3)(๐ฅ โ 3)
Factoring Special Cases
8. ๐น๐๐๐ก๐๐ 4๐ฅ2 โ 9๐ฆ2
9. ๐น๐๐๐ ๐กโ๐ ๐๐๐๐ก(๐ ) ๐ฅ2 + 12๐ฅ + 36 = 0
10. ๐น๐๐๐ก๐๐ ๐๐๐๐๐๐๐ก๐๐๐ฆ 8๐ฅ3 โ 56๐ฅ2 + 98๐ฅ
Scan this QR code to go to a video tutorial on factoring
special cases.
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Solving Quadratic Equations A.4 The student will solve multistep linear and quadratic equations in two variables
b) solving quadratic equations algebraically and graphically;
Graphing a quadratic equation
Standard form for a quadratic function is: ๐(๐ฅ) = ๐๐ฅ2 + ๐๐ฅ + ๐ , ๐ โ 0
The graph of a quadratic equation will be a parabola.
If ๐ > 0, then the parabola opens upward. If ๐ < 0, then the parabola opens downward.
The axis of symmetry is the line = โ๐
2๐ .
The x-coordinate of the vertex is โ๐
2๐ . The y-coordinate of the vertex is found by
plugging that x value into the equation and solving for ๐(๐ฅ).
The y-intercept is (0, ๐).
To graph a quadratic:
1. Identify a, b, and c.
2. Find the axis of symmetry (๐ฅ = โ๐
2๐ ), and lightly sketch.
3. Find the vertex. The x-coordinate is โ๐
2๐ . Use this to find the y-coordinate.
4. Plot the y-intercept (c), and its reflection across the axis of symmetry.
5. Draw a smooth curve through your points.
The vertex of a parabola is its turning point, or the โtipโ of the parabola. In this picture, the turning point is at (2, 0).
Example 1: Graph ๐ฆ = 2๐ฅ2 โ 4๐ฅ + 3
Step 1: Identify a, b, and c. ๐ = 2, ๐ = โ4, ๐๐๐ ๐ = 3
Step 2: Find and sketch the axis of symmetry.
๐ฅ = โ๐
2๐ ๐ฅ =
โ(โ4)
2(2) ๐ฅ =
4
4 ๐ฅ = 1
Step 3: Find the vertex.
Scan this QR code to go to a video tutorial on graphing and
solving quadratic equations.
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The x-coordinate is 1. Plug this in to find y.
๐ฆ = 2(1)2 โ 4(1) + 3 ๐ฆ = 2 โ 4 + 3 ๐ฆ = 1
The vertex is (1, 1).
Step 4: Plot the y-intercept and its reflection.
Because c = 3, the y-intercept is (0, 3). Reflecting this point across x = 1
gives the point (2, 3).
Step 5: Draw a smooth curve.
Remember to check your graphs in your calculator!
You might be asked to find the solutions of a quadratic equation by graphing it. The
solutions to a quadratic equation are the points where it crosses the x-axis.
A quadratic can have two solutions, only one solution, or no solutions at all.
Sometimes you will need to find the solution to a quadratic that cannot be factored. In
that case, you can use the quadratic formula: ๐ฅ =โ๐ยฑโ๐2โ4๐๐
2๐
You just substitute the values for a, b, and c into the quadratic formula and simplify.
Two Solutions (the parabola
crosses the x-axis twice)
One Solution (the parabola
crosses the x-axis one time)
No Solutions (the parabola does
not cross the x-axis)
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Example 2: Solve 5๐ฅ2 โ 2๐ฅ โ 9 = 0
๐ = 5 ๐ = โ2 ๐ = โ9 Plug these values into the quadratic formula
๐ฅ =โ๐ยฑโ๐2โ4๐๐
2๐ ๐ฅ =
2ยฑโ(โ2)2โ4(5)(โ9)
2(5) ๐ฅ =
2ยฑโ4+180
10 ๐ฅ =
2ยฑโ184
10
Your two solutions are ๐ฅ =2+โ184
10=
2+2โ46
10=
๐+โ๐๐
๐ and ๐ฅ =
2โโ184
10=
2+2โ46
10=
๐+โ๐๐
๐
Solving Quadratic Equations
1. Sketch the graph of ๐ฆ = ๐ฅ2 + 4
2. Sketch the graph of ๐ฆ = โ2๐ฅ2 + 6๐ฅ
3. Find the solution(s) by graphing ๐ฆ = ๐ฅ2 โ 16
4. Find the solution(s) by graphing ๐ฆ = ๐ฅ2 โ 10๐ฅ + 25
5. Find the solution(s), use the quadratic formula 3๐ฅ2 + 6๐ฅ โ 5 = 0
6. Find the zero(s) of the quadratic, use any method you like. 2๐ฅ2 = 4๐ฅ โ 9
Scan this QR code to go to a video tutorial on using the
quadratic formula.
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Answers to the
problems: Laws of Exponents
1. โ6๐4๐7
2. 25๐ฅ4๐ฆ2๐ง6
3. ๐8๐48๐24
4. 2197๐22๐8๐31
5. 3๐7
๐
6. ๐ฅ9๐ฆ3
8
7. 32๐ฅ5
๐ฆ15
8. 1
๐3
9. 5๐5๐3
3๐4
10. 9๐ง4
4๐ฅ12๐ฆ4
Polynomials
1. 5๐๐๐2 โ 7๐๐2 + 9๐๐
2. 6๐ + 8๐ โ 8๐ + 2๐ + 3๐2
3. 9๐ฅ2 โ 7๐ฅ๐ฆ + 13๐ฆ2
4. 32๐๐2 + 10๐2๐ โ 35๐2 โ ๐๐
5. 6๐ก๐ข โ 11๐ข + 2๐ก
6. โ8๐3๐3 + 6๐2๐3
7. 38๐2๐2๐ โ 13๐๐3๐ โ 31๐๐2๐2
8. 36๐ฅ2 โ 25
9. 20๐3๐2 + 5๐2๐2 โ 8๐๐3 โ 2๐3
10. 9๐ฅ2๐ฆ2 โ 30๐ฅ2๐ฆ + 25๐ฅ2 Factoring & Simplifying Radicals
1. 2 โ 2 โ 3 โ 3 โ 5 โ ๐ โ ๐
2. 3๐ฅ
3. 2(4๐3 + 7๐ โ 6)
4. 2๐ฅ2๐ฆโ๐ฆ
5. 6๐๐2๐โ15๐๐
6. 2๐ โ6๐๐23
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Factoring Special Cases
1. (๐ฅ + 4)(๐ฅ + 3)
2. 2(๐ฅ + 2๐ฆ)(๐ฅ โ 9๐ฆ)
3. (2๐ฅ + 5)(3๐ฅ + 1)
4. Prime
5. ๐ฅ = โ5, โ4 or {โ5, โ4}
6. ๐ฅ = 2,3
2 or {
3
2, 2}
7. ๐ฅ = โ6, 0, 1 or {โ6, 0, 1}
8. (2๐ฅ + 3๐ฆ)(2๐ฅ โ 3๐ฆ)
9. ๐ฅ = โ6 or {โ6}
10. 2๐ฅ (2๐ฅ โ 7)2
Solving Quadratic Equations 1.
2.
3. ๐ฅ = 4, โ4 or {โ4, 4}
4. ๐ฅ = 5 or {5}
5. โ3 ยฑ2 โ6
3
6. No Solution