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YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. |
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
| |
Completing the Square
Lessons on Quadratics: [Summary
- the Program] [ Graphing Exercises ] [ Graph y = a[(x-h)^2 +k] Theory ] [ Factoring Quadratics ] [ Difference of Two Squares ] [ Completing the Square ] [ Convert to Standard Form (Arith) ] [ Quadratic Formula ] [ Finding Coefficients ] [ Applications ] [ Quadratics Summary ] [ Exercises ]
Initial Step
Geometric Demonstration of Identity
(x+Q)2 = x2+2Qx + Q2.
valid for x and Q positive follows. Draw a square with sides of length x+Q
and then divide the sides into sub-segments of length x and Q respectively as
indicated below.
Here the equality of two different ways to calculate area of square yields
the require identity.
(x+Q)2 = x2+2Qx
+ Q2.
Column Multiplication Demonstration
x + Q
x +
Q (multiply)
x2 + Qx
Qx + Q2 (add)
x2+2Qx + Q2 |
The calculation here allows x and Q to be real numbers
without the previous restriction that they be positive. |
The identify (x+Q)2 = x2+2Qx + Q2 also
follows in general from the earlier identity
(x+A)(x+B) = x2+(A+B)x + AB
which we established earlier if we take A = B = Q in it.
2. Completing the square identity:
x2+2Qx +P= (x+Q)2 - Q2 + P
is consequence of the identity (x+Q)2 = x2+2Qx + Q2
| Thus |
The equality of two difference ways
to calculate area of square gives.
(x+Q)2 =
x2+2Qx + Q2.
Q2 = Q2
(subtract)
(x+Q)2 - Q2 = x2+2Qx |
Variation:
x2-2Qx +P = (x-Q)2 - Q2 + P
is consequence of the identity (x-Q)2 = x2 -2Qx + Q2
3. Examples
Example I
x2+6x + 5 =
= x2+2(3)x + 5 take Q = 3
= (x+3)2 - 32 + 5
= (x+3)2 - 4
Example II
x2-8x + 25 =
= x2+2(-4)x + 25 take Q = -4
= (x+-4)2 - 42 + 25
= (x+3)2 + 9
> 9 > 0
Example III
x2-8x + 12 =
= x2+2(-4)x + 12 take Q = -4
= (x+-4)2 - 42 + 12
= (x-4)2 - 4
Example IV
x2-10x + 8 =
= x2+2(-5)x + 8 take Q = -5
= (x+-5)2 - 52 + 8
= (x-5)2 - 17
Note: Completing the square may lead to a difference of squares or a
sum of squares. In the first case, how to factor the difference of squares leads
to the factorization of quadratic expressions and to the solution of quadratic
equations. See below.
Lessons on Quadratics: [ Graphing Exercises ] [ Graph y = a[(x-h)^2 +k] Theory ] [ Factoring Quadratics ] [ Difference of Two Squares ] [ Completing the Square ] [ Convert to Standard Form (Arith) ] [ Quadratic Formula ] [ Finding Coefficients ] [ Applications ] [ Quadratics Summary ] [ Exercises ]
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Analytic Geometry
Area Entrance
Entrance + Pages Below this page
Pages at Current LevelGraphing Exercises
Graph y = a[(x-h)^2 +k] Theory
Factoring Quadratics
Difference of Two Squares
Completing the Square
Convert to Standard Form (Arith)
Quadratic Formula
Finding Coefficients
Applications
Quadratics Summary
Exercises
Area Entrance
(C) Complex Numbers
(FN) What are Functions?
(FN) Functions - More
SZM: Sign Analysis
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections
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Extras: Not all perfect.
Equal Sign Use/Abuse
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
|x| Eq'ns & Inequalities
Rectangular Coords
Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
(PT) Translations
(PT) Dilatations
PT: Rotations
Links to Site Pages outside this site area follow - co- and pre-
requisites.
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Easy Consequences of this (newest) Complex
Number. Starter Lesson follow below to provide an alternate
development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
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Decimal
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Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
Simplification
Using formulas forwards
& Backwards - A unifying theme for algebra skill development - the 4th
skill in Volume 2, Three
Skills for Algebra!
What
is a Variable?
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