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YOU are better than YOU think. Show yourself how:
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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 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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-/[]\- 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. |
Quadratics
Factoring by inspection uses the equation (x+A)(x+B) = x2+(A+B)x + AB. To get the completing the square equation x2+2Qx = (x+Q)2 - Q2 take A = B = Q and then subtract Q2 from both sides. Taking B= -A gives (x+A)(x-A) = x2 - A2 or more generally, (C+A)(C-A) = C2 - A2 The latter equation provides a means to factor the difference of two squares. Memory Aid for (x+A)(x+B) = x2+(A+B)x + AB
Memory Aid for Completing the Square Identity
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| x + Q |
x2 |
Qx |
| Qx | Q2 | |
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x + Q |
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| (x+Q)2 = x2+2Qx + Q2. | ||
By completing the square, each quadratic ax2+bx+c = a[(x-q)2 + h ] with q = -b/(2a) and h = (4ac-b2)/(4a2). The graph of y = a[(x-q)2 + h ] has an axis of symmetry with equation x = q. Putting x = q gives y = aq2+bq+c = a[(q-q)2 + h ] = ah.
The point with coordinates [q, ah] = [q, aq2+bq+c] is the vertex of the quadratic. It is the lowest point on the quadratic if a> 0 and it is the highest point if a < 0. If a> 0 the quadratic opens upward. If a < 0, the quadratic opens downward.
If h < 0, then (x-q)2 + h = 0 when and only
when (x-q)2 = -h or
This gives the first way to solve a[(x-q)2 + h ] = 0 or |
If the discriminant b2-4ac > 0 then h < 0 and solutions of the quadratic equation ax2+bx+c = 0 are also given by
| x = |
2a |
Special Case: If the discriminant b2-4ac = 0 then h = 0 and the quadratic ax2+bx+c = 0 on the axis of symmetry and there is only one x-intercept, namely x = -b/(2a)
If you are given that or show that ax2+bx+c = a(x +s)(x+r) then x = -s and x = -r give one or two x-intercepts of y = ax2+bx+c, and the axis of symmetry is at x = -½(r+s) = -b/(2a), halfway between the two intercepts. You may show that show that ax2+bx+c = a(x +s)(x+r) with factoring by inspection (if it works) or via two steps: completing the square and using the difference of two squares.
One way to sketch or graph the quadratics y = ax2+bx+c or y =a[(x-q)2 + h ] is to plot points on the curve y = ax2+bx+c at the x-intercept or intercepts, if any, and for x = q, x = q ± 1/4, x = q ± 1/2, x = q ± 1, x =q ± 2, etc, and then join these points by a smooth curve. Use fewer points if time is short. Here x = q = -b/(2a) is the equation of the axis of symmetry for the curve y = ax2+bx+c. Hint: Calculate the coordinates of these points and then choose a unit lengths for the y and x axes. The unit lengths or scale on each axis may be different.
Remark: Completing the Square and the quadratic formulas works with complex numbers as well. But when were are graphing quadratics y = ax2+bx+c with x restricted to real values, complex roots are not allowed. They are extraneous. Here we have a situation where an equation ax2+bx+c = 0 may have solutions outside the set where they are meaningful.
A. (i) Use the quadratic
formula to solve x2-3x-4= 1.
(ii) Find the
value of y on the axis of symmetry of the quadratic y = x2-3x-4.
(iii) Use the
results of (i) and (ii) to sketch the curve y = x2-3x-4.
B. Find the intersection points of the quadratic y = x2 and the line y = 3x+4.
C. (i) Sketch the curve pq = 1 in
the first quadrant of the pq plane.
(ii) Give the definition
of ln(x) for x > 1.
(iii) Shade in the area
under this curve pq = 1 that gives or defines ln(4).
D. (Step I) Complete the square
for x2-6x-8 and simplify the result.
(Step II) Use the result
of step I and the difference of two squares to factor x2-6x-8
(Step III) Use the
result of step II to solve x2-6x-8 = 0
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Analytic Geometry
& Functions, etc
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 Links More Links
Pages at Above this Page
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.
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
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!
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