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Algebra Tricks and Shortcuts: A Guide to Faster Calculations, Summaries of Mathematics

Mathematics

A comprehensive guide to algebra tricks and shortcuts that can help students solve equations and perform calculations more efficiently. It covers various techniques, including factorization, distributive property, squaring shortcuts, multiplication tricks, solving linear equations, percentage and fraction calculations, binomial expansion, solving quadratic equations, and mental approximation. The document also includes a section on trigonometric functions and their graphs, providing a visual representation of their behavior.

Typology: Summaries

2024/2025

Available from 01/21/2025

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Here are some algebra tricks and shortcuts that can make calculations faster and easier:
1. Simplify Before Calculating
โ— Factorization: Break down expressions into simpler factors.
Example: Instead of calculating 48ร—2548 \times 2548ร—25, write it as
48ร—25=48ร—(100/4)=120048 \times 25 = 48 \times (100/4) =
120048ร—25=48ร—(100/4)=1200.
โ— Distributive Property:
a(b+c)=ab+aca(b + c) = ab + aca(b+c)=ab+ac
Example: 23ร—12=(20+3)ร—12=20ร—12+3ร—12=240+36=27623 \times 12 = (20 + 3) \times 12
= 20 \times 12 + 3 \times 12 = 240 + 36 =
27623ร—12=(20+3)ร—12=20ร—12+3ร—12=240+36=276.
2. Squaring Shortcuts
โ— Square of a Number Ending in 5:
n2=(nโˆ’1)(n+1)+25n^2 = (n-1)(n+1) + 25n2=(nโˆ’1)(n+1)+25
Example: 352=30ร—40+25=122535^2 = 30 \times 40 + 25 = 1225352=30ร—40+25=1225.
โ— Difference of Squares:
a2โˆ’b2=(a+b)(aโˆ’b)a^2 - b^2 = (a+b)(a-b)a2โˆ’b2=(a+b)(aโˆ’b)
Example: 1022โˆ’982=(102+98)(102โˆ’98)=200ร—4=800102^2 - 98^2 = (102 + 98)(102 - 98)
= 200 \times 4 = 8001022โˆ’982=(102+98)(102โˆ’98)=200ร—4=800.
3. Multiplication Tricks
โ— Multiply by 11:
For a two-digit number ababab, insert the sum of the digits between them.
Example: 54ร—11=5(5+4)4=59454 \times 11 = 5(5+4)4 = 59454ร—11=5(5+4)4=594.
โ— Using Base Numbers:
For numbers close to a base (10,100,etc.)(10, 100, etc.)(10,100,etc.):
Example: 97ร—96=(100โˆ’3)(100โˆ’4)=10000โˆ’300โˆ’400+12=921297 \times 96 =
(100-3)(100-4) = 10000 - 300 - 400 + 12 =
921297ร—96=(100โˆ’3)(100โˆ’4)=10000โˆ’300โˆ’400+12=9212.
4. Solving Linear Equations Quickly
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Here are some algebra tricks and shortcuts that can make calculations faster and easier:

1. Simplify Before Calculating

โ— Factorization: Break down expressions into simpler factors. Example: Instead of calculating 48ร—2548 \times 2548ร—25, write it as 48ร—25=48ร—(100/4)=120048 \times 25 = 48 \times (100/4) = 120048ร—25=48ร—(100/4)=1200. โ— Distributive Property: a(b+c)=ab+aca(b + c) = ab + aca(b+c)=ab+ac Example: 23ร—12=(20+3)ร—12=20ร—12+3ร—12=240+36=27623 \times 12 = (20 + 3) \times 12 = 20 \times 12 + 3 \times 12 = 240 + 36 = 27623ร—12=(20+3)ร—12=20ร—12+3ร—12=240+36=276.

2. Squaring Shortcuts

โ— Square of a Number Ending in 5: n2=(nโˆ’1)(n+1)+25n^2 = (n-1)(n+1) + 25n2=(nโˆ’1)(n+1)+ Example: 352=30ร—40+25=122535^2 = 30 \times 40 + 25 = 1225352=30ร—40+25=1225. โ— Difference of Squares: a2โˆ’b2=(a+b)(aโˆ’b)a^2 - b^2 = (a+b)(a-b)a2โˆ’b2=(a+b)(aโˆ’b) Example: 1022โˆ’982=(102+98)(102โˆ’98)=200ร—4=800102^2 - 98^2 = (102 + 98)(102 - 98) = 200 \times 4 = 8001022โˆ’982=(102+98)(102โˆ’98)=200ร—4=800.

3. Multiplication Tricks

โ— Multiply by 11: For a two-digit number ababab, insert the sum of the digits between them. Example: 54ร—11=5(5+4)4=59454 \times 11 = 5(5+4)4 = 59454ร—11=5(5+4)4=594. โ— Using Base Numbers: For numbers close to a base (10,100,etc.)(10, 100, etc.)(10,100,etc.): Example: 97ร—96=(100โˆ’3)(100โˆ’4)=10000โˆ’300โˆ’400+12=921297 \times 96 = (100-3)(100-4) = 10000 - 300 - 400 + 12 = 921297ร—96=(100โˆ’3)(100โˆ’4)=10000โˆ’300โˆ’400+12=9212.

4. Solving Linear Equations Quickly

โ— Cross Multiplication for Proportions: If ab=cd\frac{a}{b} = \frac{c}{d}ba=dc, then ad=bcad = bcad=bc. โ— Shortcut for Two Equations: ax+by=cax + by = cax+by=c dx+ey=fdx + ey = fdx+ey=f Use elimination by multiplying equations to make xxx or yyy coefficients equal.

5. Percentage and Fractions

โ— Quick Percent Calculation: 18%18%18% of 505050: Rewrite as (50ร—0.18)=50ร—(0.2โˆ’0.02)=10โˆ’1=9(50 \times 0.18) = 50 \times (0.2 - 0.02) = 10 - 1 = 9(50ร—0.18)=50ร—(0.2โˆ’0.02)=10โˆ’1=9. โ— Convert Difficult Fractions to Easier Values: 1316\frac{13}{16}1613 as 0.81250.81250.8125: Estimate it as 1216=0.75\frac{12}{16} = 0.751612=0.75.

6. Binomial Expansion (Shortened)

For (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2, use direct substitution: Example: (3x+2)2=(3x)2+2(3x)(2)+22=9x2+12x+4(3x + 2)^2 = (3x)^2 + 2(3x)(2) + 2^2 = 9x^2 + 12x + 4(3x+2)2=(3x)2+2(3x)(2)+22=9x2+12x+4.

7. Solving Quadratic Equations

Use the quadratic formula x=โˆ’bยฑb2โˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2aโˆ’bยฑb2โˆ’4ac, but memorize shortcuts like:

โ— If b=0b = 0b=0: x=ยฑโˆ’c/ax = \pm \sqrt{-c/a}x=ยฑโˆ’c/a. โ— Factorize directly when possible: Example: x2โˆ’5x+6=0x^2 - 5x + 6 = 0x2โˆ’5x+6=0 becomes (xโˆ’2)(xโˆ’3)=0(x-2)(x-3) = 0(xโˆ’2)(xโˆ’3)=0, so x=2,3x = 2, 3x=2,3.

8. Mental Approximation

โ— Round off to make numbers easier to handle, then adjust the final result. Example: 398ร—52โ‰ˆ400ร—50=20000398 \times 52 \approx 400 \times 50 =

  1. Exponential or Logarithmic Functions : For example, y=exy = e^xy=ex or y=log(x)y = \log(x)y=log(x).
  2. Trigonometric Functions : For example, y=sin(x)y = \sin(x)y=sin(x), y=cos(x)y = \cos(x)y=cos(x), etc. โ—‹ Periodic wave-like graphs.
  3. Sine (sin): The blue wave-like curve oscillates between -1 and 1.
  4. Cosine (cos): The green wave-like curve, similar to sine but phase-shifted.
  5. Tangent (tan): The red curve with periodic vertical asymptotes at ฯ€2+nฯ€\frac{\pi}{2} + n\pi2ฯ€+nฯ€, limited to avoid extremes.

Customized Graph

  1. Add More Trigonometric Functions: Include secant, cosecant, or cotangent.
  2. Change Range: Adjust the xxx-axis range (e.g., zoom in or out).
  3. Change Colors or Styles: Use different colors, line styles, or markers.
  4. Highlight Specific Points: Mark key points like maxima, minima, or intercepts.
  5. Add Gridlines or Labels: Include custom labels for specific angles like ฯ€/2,ฯ€,2ฯ€\pi/2, \pi, 2\piฯ€/2,ฯ€,2ฯ€, etc.
  6. Focus on a Single Function: Plot only one function, like sine or cosine

The graph has been customized with gridlines and labeled key points:

  1. Gridlines: Vertical dashed lines at key angles like โˆ’2ฯ€,โˆ’ฯ€,0,ฯ€,2ฯ€-2\pi, -\pi, 0, \pi, 2\piโˆ’2ฯ€,โˆ’ฯ€,0,ฯ€,2ฯ€.
  2. Labeled Axes: The xxx-axis labels include specific points like ฯ€2\frac{\pi}{2}2ฯ€ and ฯ€\piฯ€, displayed in their mathematical notation.