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YOU are better than YOU think. Show yourself how: |
-/[]\- Logic
Mastery Do not leave here without it - 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 linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; 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. |
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6. 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
used in factoring by inspection
| x + A |
x2 |
Bx |
| Ax | AB | |
|
x + B |
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| For x, A and B all positive, the area of the
large rectangle is (x+A)(x+B) or the sum of the areas of the small
rectangle. This implies (x+A)(x+B) = x2+(A+B)x + AB.
The condition that x, A and B all be positive can be removed if one
uses the distributive law twice to obtain this result (x+A)(x+B) = x(x+B) + A(x+B)
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Memory Aid for Completing the Square Identity
x2+2Qx = (x+Q)2 - Q2
| x + Q |
x2 |
Qx |
| Qx | Q2 | |
|
x + Q |
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| (x+Q)2 = x2+2Qx + Q2. | ||
Quadratic Formula and Related Material
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.
Graphing Quadratics
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.
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In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.