Thinking Mathematically

Thinking Mathematically Algebra 1 By: A.J. Mueller

Properties

Proprieties • Addition Property (of Equality) • 4+5=9 • Multiplication Property (of Equality) • 5●8=40 • Reflexive Property (of Equality) • 12=12 • Symmetric Property (of Equality) • If a=b then b=a

Proprieties • Transitive Property (of Equality) • If a=b and b=c then a=c • Associative Property of Addition • (0.6+5.3)+4.7=0.6+(5.3+4.7) • Associative Property of Multiplication • (-5●7) 3=-5(7●3) • Commutative Property of Addition • 2+x=x+2

Proprieties • Communicative Property of Multiplication • b3a2=a2b3 • Distributive Property • 5(2x+7)= 10x+35 • Prop. of Opposites or Inverse Property of Addition • a+(-a)=0 and (-a)+a=0 • Prop. of Reciprocals or Inverse Prop. of Multiplication • x2/7•7/x2=1

Proprieties • Identity Property of Addition • -5+0=-5 • Identity Property of Multiplication • x●1=x • Multiplicative Property of Zero • 5●0=0 • Closure Property of Addition • For real a and b, a+b is a real number

Proprieties • Closure Property of Multiplication • ab = ba • Product of Powers Property • x3+x4=x7 • Power of a Product Property • (pq)7=p7q7 • Power of a Power Property • (n2) 3

Proprieties • Quotient of Powers Property • X5/x3=x2 • Power of a Quotient Property • (a/b) 2 • Zero Power Property • (9ab)0=1 • Negative Power Property • h-2=1/h2

Proprieties • Zero Product Property • ab=0, then a=0 or b=0 • Product of Roots Property • √20= √4•√5 • Power of a Root Property • (√7) 2=7

Solving 1st Power Inequalities in One Variable

-5 Solving 1st Power Inequalities in One Variable With only one inequality sign x > -5 Solution Set: {x: x > -5} Graph of the Solution:

9 -4 Conjunctions • Open endpoint for these symbols: > < • Closed endpoint for these symbols: ≥ or≤ • Conjunction must satisfy both conditions • Conjunction = “AND” {x: -4 < x ≤ 9}

7 -4 Disjunctions • Open endpoint for these symbols: > < • Closed endpoint for these symbols: ≥ or ≤ • Disjunction must satisfy either one or both of the conditions • Disjunction = “OR” {x: x < -4 or x ≥ 7}

1 -5 Special Cases That = {All Reals} • Watch for special cases • No solutions that work: Answer is Ø • Every number works: Answer is {reals} • Disjunction in same direction: answer is one arrow {x: x > -5 or x ≥ 1}

Ø Special Cases That = {x: -x < -2 and -5x ≥ 15}

Linear equations in two variables

Linear equations in two variables • Lots to cover here: slopes of all types of lines; equations of all types of lines, standard/general form, point-slope form, how to graph, how to find intercepts, how and when to use the point-slope formula, etc. Remember you can make lovely graphs in Geometer's Sketchpad and copy and paste them into PPT.

Important Formulas • Slope- • Standard/General form- ax+bx=c • Point-slope form- • Use point-slope formula when you know 4 points on 2 lines. • Vertex- • X-intercepts- set f(x) to 0 then solve • Y-intercepts- set the x in the f(x) to 0 and then solve

Examples of Linear Equations • Example 1 • y=-3/4x-1

Examples of Linear Equations • Example 2 • 3x-2y=6 (Put into standard form) • 2y=-3x+6 (Divide by 2) • y=-3/2x+6 (Then graph)

Linear Systems

Substitution Method • Goal: replace one variablewith an equal expression Step 1: Look for a variable with a coefficient of one. Step 2: Isolate that variable Equation A now becomes: y = 3x + 1 Step 3: SUBSTITUTE this expression into that variable in Equation B Equation B now becomes 7x – 2( 3x + 1 ) = - 4 Step 4: Solve for the remaining variable Step 5: Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!)

Addition/ Subtraction (Elimination) Method • Goal: Combine equations to cancel out one variable. Step 1: Look for the LCM of the coefficients on either x or y. (Opposite signs are recommended to avoid errors.) Here: -3y and +2y could be turned into -6y and +6y Step 2: Multiply each equation by the necessary factor. Equation A now becomes: 10x – 6y = 10 Equation B now becomes: 9x + 6y = -48 Step 3: ADD the two equations if using opposite signs (if not, subtract) Step 4: Solve for the remaining variable Step 5: Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.)

Factoring

Types of Factoring • Greatest Common Factor (GFC) • Difference of Squares • Sun and Difference of Cubes • Reverse FOIL • Perfect Square Trinomial • Factoring by Grouping (3x1 and 2x2)

GFC • To find the GCF, you just look for the variable or number each of the numbers have in common. • Example 1 • x+25x+15 • x(25+15)

Difference of Squares • Example 1 • 27x4+75y4 • 3(9x4+25y4) • 3(3x2+5y2)(3x2-5y2) • Example 2 • 45x6-81y4 • 9(5x4-9y4)

Sun and Difference of Cubes • Example 1 • (8x3+27) • (2x+3) • (4x2-6x+9) • Example 2 • (p3-q3) • (p-q) (p2+pq+q2)

Reverse FOIL • Example 1 • x2-19x-32 • (x+8)(x-4) • Example 2 • 6y2-15y+12 • (3y-4)(2y-3)

Perfect Square Trinomial • Example 1 • 4y2+30y+25 • (2y+5) 2 • Example 2 • x2-10x+25 • (x-5) 2

Factoring By Grouping • 3x1 • Example 1 • a2+4a+4-b2 • (a+4a+4)-(b2) • (a+2)-(b2) • (a+2-b)(a+2+b)

Factoring by Grouping • 2x2 • Example 1 • 2x+y2+4x+4y • [x+y][2+y]+4[x+y]

Quadratic Equations

Factoring Method • Set equal to zero • Factor • Use the Zero Product Property to solve. • Each variable equal to zero.

Factoring Method Examples • Any # of terms- look for GCF first • Example 1 • 2x2=8x (subtract 8x to set equation equal to zero) • 2x2-8x=0 (now factor out the GCF) • 2x(x-4)=0

Factoring Method Examples • Set 2x=0, divide 2 on both sides and x=0 • Set x-4=0, add 4 to both sides and x=4 • x is equal to 0 or 4 • The answer is {0,4}

Factoring Method- Binomials • Binomials – Look for Difference of Squares • Example 1 • x2=81 (subtract 81 from both sides) • x2-81=0 (factoring equation into conjugates) • (x+9)(x-9)=0 • x+9=0 or x-9=0

Factoring Method- Binomials • x+9=0 (subtract 9 from both sides) • x=-9 • x-9=0 (add 9 to both sides) • x=9 • The answer is {-9,9}

Factoring Method-Trinomials • Trinomials – Look for PST • Example 1 • x2-9x=-18 (add 18 to both sides) • x2-9x+18=0 (x2-9x+18 is a PST) • (x-9)(x-9)=0 • x-9=0 (add 9 to both sides) x=9 • The answer is {9d.r.} d.r.- double root

Square Roots of Both Sides • Reorder terms IF needed • Works whenever form is (glob)2 = c • Take square roots of both sides • Simplify the square root if needed • Solve for x, or in other words isolate x.

Square Roots Of Both Sides • Example 1 • (Factor out the GCF) • 2(x2-6x-2)=0 (You can get rid of the 2 because it does not play a role in this type of equation) • x2-6-2x=0 (Add the 2 to both sides) • x2-6x__=2__ (Take half of the middle number which right now is 6) • x2-6x+9=2+9 (Simplify)

Square Roots Of Both Sides • (x-3)=11 (Then take the square root of both sides) • (x-3)= 11 (Continue to simplifying) • (Add the 3 to both sides) • (Final Answer)

Completing the Square • Example 1 • 2x2-12x-4=0 (Factor out the GCF) • 2(x2-6x-2)=0 (You can get rid of the 2 because it does not play a role in this type of equation) • x2-6-2x=0 (Add the 2 to both sides) • x2-6x__=2__ (Take half of the middle number which right now is 6) • x2-6x+9=2+9 (Simplify)

Completing the Square • (x-3)=11 (Then take the square root of both sides) • √(x-3)= +/-√11 (Continue to simplifying) • (x-3)=+/- √11 (Add the 3 to both sides) • x=3+/- √11 (Final Answer)

Quadratic Formula • This is a formula you will need to memorize! • Works to solve all quadratic equations • Rewrite in standard form in order to identify the values of a, b and c. • Plug a, b & c into the formula and simplify! • QUADRATIC FORMULA:

Quadratic Formula Examples • Example 1 • 3x2-6=x2+12x • Put this in standard form: 2x2-12x-6=0 • Put into quadratic formula

Quadratic Formula Examples

The Discriminant – Making Predictions • b2-4ac2 is called the discriminant • Four Cases 1. b2 – 4acpositive non-square two irrational roots 2. b2 – 4acpositive square two rational roots 3. b2 – 4aczero one rational double root 4. b2 – 4acnegative no real roots

The Discriminant – Making Predictions Use the discriminant to predict how many “roots” each equation will have. 1. x2 – 7x – 2 = 0 49–4(1)(-2)=57 2 irrational roots 2. 0 = 2x2– 3x + 1 9–4(2)(1)=1  2 rational roots 3. 0 = 5x2 – 2x + 3 4–4(5)(3)=-56  no real roots 4. x2– 10x + 25=0 100–4(1)(25)=0  1 rational double root

The Discriminant – Making Predictions The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have and where the vertex is located. 1. y = 2x2 – x - 6 1–4(2)(-6)=49  2 rational zeros opens up/vertex below x-axis/2 x-intercepts 2. f(x) = 2x2 – x + 6 1–4(2)(6)=-47  no real zeros opens up/vertex above x-axis/No x-intercepts

Thinking Mathematically