Essential Algebra Shortcuts: Your Quick Reference Guide

Cheatsheets

Multiplication Shortcuts

Multiplying by Powers of 10

ShortcutExample
To multiply by 10, add a zero45 × 10 = 450
To multiply by 100, add two zeros45 × 100 = 4,500
To multiply by 1000, add three zeros45 × 1000 = 45,000

Multiplying by 5

ShortcutExample
Multiply by 10, then divide by 226 × 5 = (26 × 10) ÷ 2 = 260 ÷ 2 = 130

Multiplying by 9

ShortcutExample
Multiply by 10, then subtract the original number34 × 9 = (34 × 10) – 34 = 340 – 34 = 306

Multiplying by 11

ShortcutExample
For single-digit numbers: Repeat the digit4 × 11 = 44
For two-digit numbers: Add the digits and place between them25 × 11 = 2(2+5)5 = 275
Note for two-digit numbers: If sum exceeds 9, carry the 185 × 11 = 8(8+5)5 = 8(13)5 = 935

Multiplying by 25

ShortcutExample
Multiply by 100, then divide by 432 × 25 = (32 × 100) ÷ 4 = 3200 ÷ 4 = 800

Squaring Numbers Ending in 5

ShortcutExample
Take the tens digit, multiply by next number, append 2535² = 3 × 4 = 12, then 12 + “25” = 1225
For 85²: 8 × 9 = 72, then 722585² = 7225

Multiplying Numbers with Same Tens Digit and Ones Digits Sum to 10

ShortcutExample
(Tens digit)(Tens digit + 1) + (Ones digit × Ones digit′)76 × 74 = 7 × 8 + 6 × 4 = 56 + 24 = 5624
For 76 × 74: 7 × 8 = 56, 6 × 4 = 24, so 562476 × 74 = 5624

Difference of Squares

ShortcutExample
a² – b² = (a+b)(a-b)33² – 27² = (33+27)(33-27) = 60 × 6 = 360

Square of Sum

ShortcutExample
(a+b)² = a² + 2ab + b²(30+4)² = 30² + 2(30)(4) + 4² = 900 + 240 + 16 = 1156

Square of Difference

ShortcutExample
(a-b)² = a² – 2ab + b²(30-4)² = 30² – 2(30)(4) + 4² = 900 – 240 + 16 = 676

Division Shortcuts

Dividing by Powers of 10

ShortcutExample
To divide by 10, move decimal point 1 place left450 ÷ 10 = 45.0 = 45
To divide by 100, move decimal point 2 places left4500 ÷ 100 = 45.00 = 45

Dividing by 5

ShortcutExample
Multiply by 2, then divide by 1085 ÷ 5 = (85 × 2) ÷ 10 = 170 ÷ 10 = 17

Dividing by 25

ShortcutExample
Multiply by 4, then divide by 100375 ÷ 25 = (375 × 4) ÷ 100 = 1500 ÷ 100 = 15

Dividing by 9

ShortcutExample
Use decimal pattern: 1/9 = 0.111…7 ÷ 9 = 7 × (1/9) = 7 × 0.111… = 0.777…

Factoring Shortcuts

Detecting Factors Quickly

NumberDivisibility Test
2Last digit is 0, 2, 4, 6, or 8
3Sum of digits is divisible by 3
4Last two digits form a number divisible by 4
5Last digit is 0 or 5
6Divisible by both 2 and 3
8Last three digits form a number divisible by 8
9Sum of digits is divisible by 9
10Last digit is 0
11Difference of (sum of digits in odd positions) and (sum of digits in even positions) is divisible by 11

Factoring Trinomials (ax² + bx + c)

When a = 1 (x² + bx + c)

  1. Find two numbers that:
    • Multiply to give c
    • Add to give b
  2. Use these numbers to factor: x² + bx + c = (x + m)(x + n) where m × n = c and m + n = b

Example: x² + 7x + 12

  • Factors of 12 that add to 7: 3 and 4
  • Solution: (x + 3)(x + 4)

When coefficient of x² is not 1 (ax² + bx + c)

Shortcut: Look for factors of a × c that add to b

Example: 3x² + 14x + 15

  1. a × c = 3 × 15 = 45
  2. Factors of 45 that add to 14: 9 and 5
  3. Rewrite middle term: 3x² + 9x + 5x + 15
  4. Group terms: 3x(x + 3) + 5(x + 3)
  5. Factor out (x + 3): (x + 3)(3x + 5)

Quick Factoring Patterns

ExpressionFactorizationExample
a² + 2ab + b²(a + b)²x² + 6x + 9 = (x + 3)²
a² – 2ab + b²(a – b)²x² – 10x + 25 = (x – 5)²
a² – b²(a + b)(a – b)49 – x² = (7 + x)(7 – x)
a³ + b³(a + b)(a² – ab + b²)x³ + 8 = (x + 2)(x² – 2x + 4)
a³ – b³(a – b)(a² + ab + b²)x³ – 27 = (x – 3)(x² + 3x + 9)

Equation-Solving Shortcuts

Solving Linear Equations

ShortcutExample
Combine like terms first7x + 3 – 2x – 5 = 11 → 5x – 2 = 11
Isolate variable terms on one side, constants on the other5x – 2 = 11 → 5x = 13 → x = 13/5
Work with fractions? Multiply all terms by the LCD(x/3) + (x/2) = 5 → Multiply by 6 → 2x + 3x = 30 → 5x = 30 → x = 6

Solving Quadratic Equations

Perfect Square Trinomials

ShortcutExample
Identify a² + 2ab + b² or a² – 2ab + b² patternx² + 12x + 36 = 0 → (x + 6)² = 0 → x = -6

Difference of Squares

ShortcutExample
For ax² – b = 0, solve using square root property4x² – 25 = 0 → 4x² = 25 → x² = 25/4 → x = ±5/2

Factoring

ShortcutExample
Factor and set each factor equal to zerox² – 3x – 10 = 0 → (x – 5)(x + 2) = 0 → x = 5 or x = -2

Quadratic Formula Quick Application

Type of QuadraticShortcut
If b² >> 4acThe smaller root is approximately c/b
If 4ac >> b²Roots are approximately ±√(c/a)
If b is evenSimplify to x = (-b/2 ± √((b/2)² – ac))/a

Algebraic Expression Shortcuts

Simplifying Expressions

Distributive Property Shortcuts

ShortcutExample
a(b + c) = ab + ac5(2x + 3) = 10x + 15
a(b – c) = ab – ac7(3y – 4) = 21y – 28

FOIL Method Alternative

For (a + b)(c + d), think “First, Outer, Inner, Last” or use pattern:

ShortcutExample
(a + b)(c + d) = ac + ad + bc + bd(x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12

Expanding (a + b)² Quickly

ShortcutExample
(a + b)² = a² + 2ab + b²(x + 5)² = x² + 2(5x) + 5² = x² + 10x + 25

Expanding (a – b)² Quickly

ShortcutExample
(a – b)² = a² – 2ab + b²(x – 7)² = x² – 2(7x) + 7² = x² – 14x + 49

Fraction Shortcuts

Adding Fractions with Same Denominator

ShortcutExample
Just add numerators, keep denominator(3/8) + (5/8) = (3+5)/8 = 8/8 = 1

Adding Fractions with Different Denominators

ShortcutExample
For denominators 2, 4, 8, 16… double and halve(1/2) + (3/8) = (4/8) + (3/8) = 7/8
For patterned denominators, use LCD(1/3) + (1/6) = (2/6) + (1/6) = 3/6 = 1/2

Multiplying Fractions

ShortcutExample
Cancel common factors before multiplying(2/3) × (9/10) = (2/3) × (9/10) = (2×9)/(3×10) = 18/30 = 3/5
Alternatively: Cancel 2 and 10, 3 and 9(2/3) × (9/10) = (1/3) × (3/5) = 3/15 = 1/5

Dividing Fractions

ShortcutExample
“Keep, Change, Flip”(3/4) ÷ (2/5) = (3/4) × (5/2) = (3×5)/(4×2) = 15/8

Exponent and Root Shortcuts

Exponent Laws Quick Reference

LawShortcutExample
Product Rulea^m × a^n = a^(m+n)2³ × 2⁴ = 2⁷ = 128
Quotient Rulea^m ÷ a^n = a^(m-n)5⁶ ÷ 5⁴ = 5² = 25
Power Rule(a^m)^n = a^(m×n)(3²)⁴ = 3⁸ = 6,561
Negative Exponentsa^(-n) = 1/(a^n)2⁻³ = 1/2³ = 1/8 = 0.125

Quick Mental Square Roots

NumberSquare RootNumberSquare Root
11366
42497
93648
164819
25510010

Square Root Estimation

RangeEstimate FormulaExample
Between perfect squaresUse interpolation√50 is between √49 = 7 and √64 = 8, closer to 7, so ≈ 7.07
Close to perfect square√(n² ± a) ≈ n ± a/(2n)√24 = √(25 – 1) ≈ 5 – 1/(2×5) = 5 – 0.1 = 4.9

Logarithm Shortcuts

Logarithm Properties for Quick Calculations

PropertyShortcutExample
log(1) = 0Any baselog₂(1) = 0
log₂(2) = 1Base matches argumentlog₅(5) = 1
log(ab) = log(a) + log(b)Convert products to sumslog(8×4) = log(8) + log(4)
log(a/b) = log(a) – log(b)Convert divisions to differenceslog(50/2) = log(50) – log(2)
log(a^n) = n·log(a)Pull out exponentslog(5³) = 3·log(5)

Base Conversion

ShortcutExample
log_b(x) = log(x) / log(b)log₂(16) = log(16) / log(2) = 4

Problem Solving Shortcuts

Word Problem Structures

Problem TypeVariable Assignment Shortcut
Age problemsPresent age = x, Future/Past = x ± years
Rate problemsRate × Time = Distance
Mixture problems(Amount × Concentration) of each component = Total amount × Final concentration

System of Equations Shortcuts

Elimination Method Shortcut

When coefficients are multiples:

  1. Multiply one equation so coefficients of one variable match
  2. Add or subtract equations to eliminate variable
  3. Solve for remaining variable

Example:

2x + 3y = 8
4x - 5y = 7

Multiply first equation by 2:

4x + 6y = 16
4x - 5y = 7

Subtract second from first:

11y = 9
y = 9/11

Substitution Method Shortcut

For equations where one variable has coefficient 1:

  1. Solve that equation for the variable
  2. Substitute into the other equation

Example:

x + 2y = 10
3x - y = 5

From first equation: x = 10 – 2y Substitute into second:

3(10 - 2y) - y = 5
30 - 6y - y = 5
30 - 7y = 5
-7y = -25
y = 25/7

Quadratic Applications Shortcuts

Maximum/Minimum Values

For a quadratic f(x) = ax² + bx + c:

  • Vertex occurs at x = -b/(2a)
  • Maximum value (if a < 0) or minimum value (if a > 0) is f(-b/(2a))

Quadratic Word Problems

Problem TypeShortcut
Area problemsIf length and width sum to p and area is A, dimensions are (p/2) ± √((p/2)² – A)
Projectile motionFor h = -16t² + vt + h₀, max height at t = v/32

Mental Math Shortcuts

Powers of 2

PowerValuePowerValue
2¹22⁶64
2²42⁷128
2³82⁸256
2⁴162⁹512
2⁵322¹⁰1024

Decimal to Fraction Conversions

DecimalFractionDecimalFraction
0.251/40.3753/8
0.51/20.6255/8
0.753/40.8757/8
0.11/100.333…1/3
0.21/50.666…2/3

Percentage Shortcuts

OperationShortcutExample
Find 10%Divide by 1010% of 230 = 23
Find 1%Divide by 1001% of 230 = 2.3
Find 5%Halve 10%5% of 230 = 11.5
Find 20%Double 10%20% of 230 = 46
Find 25%Divide by 425% of 230 = 57.5
Find 50%Divide by 250% of 230 = 115

Common Pitfalls and Error Prevention

Order of Operations Reminders

  • Exponents before multiplication/division
  • Multiplication/division before addition/subtraction
  • Leftmost operation first when operations have equal precedence

Sign Rule Reminders

OperationRuleExample
Negative × NegativePositive(-5) × (-3) = 15
Negative × PositiveNegative(-5) × 3 = -15
Negative + NegativeMore negative(-5) + (-3) = -8
Negative – NegativeThink addition(-5) – (-3) = (-5) + 3 = -2

Factoring Checks

  • Always check your answer by multiplying the factors back
  • When solving quadratic equations, check solutions in original equation

Equation Solving Verification

After solving x = [value], substitute back into original equation to verify

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