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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.
| |
Factoring Quadratic Expressions
arithmetic examples - working with numbers
The quadratic formula for finding roots of expressions comes follow
from (i) completing the square and then (ii) factoring if (i) results in the
difference of two squares. Numerical examples follow to illustrates the step.
The quadratic formula itself is derived in the next lesson.
In other parts of this site (see chapters 14 an 15 of the
site volume 2, Three
Skills for Algebra, arithmetic and algebraic solutions to questions have
been developed. This lesson provides the arithmetic solution for solving
quadratics equations or factoring them. The next lesson gives the algebraic
solution or route. In the next lesson, the general case in which the letters
a, b and c are not immediately replaced by numbers follows the same route and
leads to the quadratic formula.
1. Quadratic with two roots
x2-8x + 12 =
= x2+2(-4)x + 12 take Q = -4
= (x+-4)2 - 42 + 12
= (x-4)2 - 4
= (x-4)2 - 22 = a difference of two squares
= (x -4 +2) (x -4 -2)
= (x -2)(x-6)
Now x2-8x + 12 = (x -2)(x - 6) = 0 when and only when x = 2 or x =
6 by the zero product law.
Exercise: Identify the intervals where the factors (x--2) and (x-6)
are postive and negative. Then do a sign analysis of the product
y = (x -2)(x-6) = x2-8x + 12
The format for this sign analysis appears in Chapter 3, Slope
Sign Analysis, of Volume 3, Why Slopes and More Math.
Remark: Factoring by inspection would be quicker here as -8 = -2 + -6
and 12 =( 6)(1). But the aim here was to illustrate the route: Complete
the square and then factor, if possible, using the difference of two squares.
2. Quadratic with two roots
x2+6x + 5 =
= x2+2(3)x + 5 take Q = 3
= (x+3)2 - 32 + 5
= (x+3)2 - 4
= (x+3)2 - 22
= a difference of two squares
Factoring the difference of two squares (x+3)2 - 22 will
tell us when x2+6x + 5 may have the value zero,
0 = x2+6x + 5
= (x+3)2 - 22
= (x+3+2)(x+3-2)
= (x+5)(x+1)
The zero product law says the latter product is zero when and only when
x + 5 = 0 or x+ 1 = 0.
That is, when and only when
x = -5 or x = -1
respectively.
Exercise: Identify the intervals where the factors (x+5) and (x+1) are
postive and negative. Then do a sign analysis of the product.
y = x2+6x + 5 = (x+5)(x+1)
The format for this sign analysis appears in Chapter 3, Slope
Sign Analysis, of Volume 3, Why Slopes and More Math.
Remark: Factoring by inspection would be quicker here as 6 = 5 + 1 and 5
= (5)(1). But the aim here was to illustrate the route: Complete the
square and then factor, if possible, using the difference of two squares.
3. Quadratic with no roots
x2-8x + 25 =
= x2+2(-4)x + 25 take Q = -4
= (x+-4)2 - 42 + 25
= (x+3)2 + 9
= (x+3)2 + 32 = a sum of two squares
> 9
> 0
So x2-8x + 25 > 9 is never zero
Exercise: Identify the interval where (x+3)2 + 9 is
positive.
Optional Exercise: Read about complex
numbers and then write 9 = - (3 i)2 to obtain a difference
of two squares. Here i = sqrt(-1)
4. Quadratics with coefficient a of x2 not unity (not 1)
3x2- 24x + 10 =
= 3 [x2-8x + 12]
= 3[x2+2(4)x - 42 +12] take Q = 4
= 3[(x+4)2 - 4]
5. Using q = [sqrt(q)]2 in factoring a quadratic
Example: Factor x2-10x + 8
x2-10x + 8 =
= x2+2(-5)x + 8 take Q = -5
= (x+-5)2 - 52 + 8
= (x-5)2 - 17
= (x-5)2 - [(17)½]2
= a difference of two squares, namely the square of x -5 and the square
17 of the square root (17)½
Factoring the difference of two squares will tell us when
(x-5)2 - [(17)½]2
may have the value zero:
Here: (x-5)2 - [(17)½]2
= [(x-5) + (17)½]
[(x-5) - (17)½]
= [(x-5) + (17)½] [(x-5)
- (17)½]
= [ x - {5 - (17)½}]
[ x - {5 + (17)½]} ]
The Zero Product law implies the latter can be zero when and only when
x = 5 - (17)½
or x = 5 + (17)½
Use a calculator to find the approximate decimal values to locate these roots
on the x-axis.
Exercise: Identify the intervals where the factors (x--2) and (x-6)
are postive and negative. Then do a sign analysis of the product
y = x2-10x + 8 = [ x - {5
- (17)½}] [ x - {5 + (17)½]}
]
The format for this sign analysis appears in Chapter 3, Slope
Sign Analysis, of Volume 3, Why Slopes and More Math.
Remark: Factoring by inspection would be quicker here as -8 = -2 + -6
and 12 =( 6)(1). But the aim here was to illustrate the route: Complete
the square and then factor, if possible, using the difference of two squares.
Remark: Any number q > 0 equals to the square of
its square root. That is, = [sqrt(q)]2 allows to write 17 as as
[(17)½]2 in the foregoing.
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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
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Extras: Not all perfect.
Equal Sign Use/Abuse
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
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Shortest Path
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Add & Multiply Points
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Radians
(A) Vectors
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(A) Addition Geometrically
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PT: Rotations
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Number. Starter Lesson follow below to provide an alternate
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Vec & Cmplx No Applet
B2 C. Conjugates
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Divisors
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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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