The Field Manual

The 20+ formulas they don't give you. Detailed evidence of why traditional methods are too slow.

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# Concept Formula Mastery Bonus Done
Geometry

1. Circle Equation

\[ (x-h)^2 + (y-k)^2 = r^2 \]

Identify circle centers instantly.

"Find the center of \(x^2 + y^2 - 6x + 8y = 0\)."

🚫 The School Way (90+ Seconds)
  • Step 1: Start with \(x^2 + y^2 - 6x + 8y = 0\).
  • Step 2: Group x terms: \((x^2 - 6x) + (y^2 + 8y) = 0\).
  • Step 3: Complete square for x: Half of -6 is -3, square is 9.
  • Step 4: Add/subtract 9: \((x^2 - 6x + 9 - 9) + (y^2 + 8y) = 0\).
  • Step 5: Complete square for y: Half of 8 is 4, square is 16.
  • Step 6: Add/subtract 16: \((x^2 - 6x + 9) + (y^2 + 8y + 16 - 16) - 9 = 0\).
  • Step 7: Move constants right: \((x^2 - 6x + 9) + (y^2 + 8y + 16) = 9 + 16\).
  • Step 8: Factor squares: \((x - 3)^2 + (y + 4)^2 = 25\).
  • Step 9: Identify center from \((x - h)^2 + (y - k)^2 = r^2\).
  • Step 10: Center is (3, -4).
  • Double the work, double the mistakes!
✅ The Practix Way
  • Step 1: Half of -(-6) = 3.
  • Step 2: Half of -(8) = -4.
  • Center: (3, -4).
Practix Secret
Quadratics

2. The Vertex Shortcut

\[ x = -\frac{b}{2a} \]

Instantly find the maximum or minimum of any parabola.

"Find the x-coordinate of the vertex for \(y = 3x^2 - 12x + 7\)."

🚫 The School Way (70+ Seconds)
  • Step 1: Look at equation \(y = 3x^2 - 12x + 7\). Need to complete the square.
  • Step 2: Factor out coefficient of \(x^2\): \(y = 3(x^2 - 4x) + 7\).
  • Step 3: Inside parentheses, identify b = -4.
  • Step 4: Take half of b: \(-4 \div 2 = -2\).
  • Step 5: Square that result: \((-2)^2 = 4\).
  • Step 6: Add AND subtract inside: \(3(x^2 - 4x + 4 - 4) + 7\).
  • Step 7: Separate the -4: \(3(x^2 - 4x + 4) + 3(-4) + 7\).
  • Step 8: Distribute the 3: \(3(x^2 - 4x + 4) - 12 + 7\).
  • Step 9: Factor perfect square: \(3(x - 2)^2 - 12 + 7\).
  • Step 10: Simplify constants: \(3(x - 2)^2 - 5\).
  • Step 11: Vertex form is \(a(x-h)^2 + k\), so h = 2.
  • One wrong sign ruins everything!
✅ The Practix Way (2s)
  • Step 1: \(x = -(-12) / (2 \times 3)\)
  • Step 2: \(x = 12 / 6 = 2\)
Quadratics

3. Product of Solutions

\[ \text{Product} = \frac{c}{a} \]

Find the product of all x-values without solving.

"Product of solutions for \(4x^2 - 19x - 12 = 0\)."

🚫 The School Way (65+ Seconds)
  • Step 1: Identify coefficients: \(a=4, b=-19, c=-12\).
  • Step 2: Write Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).
  • Step 3: Calculate \(b^2\): \((-19)^2 = 361\).
  • Step 4: Calculate \(4ac\): \(4(4)(-12) = -192\).
  • Step 5: Calculate discriminant: \(361 - (-192) = 361 + 192 = 553\).
  • Step 6: Find \(\sqrt{553}\): Calculator shows ≈ 23.516.
  • Step 7: Plug into formula: \(x = \frac{19 \pm 23.516}{8}\).
  • Step 8: Find first root: \(x_1 = \frac{19 + 23.516}{8} = \frac{42.516}{8} = 5.3145\).
  • Step 9: Find second root: \(x_2 = \frac{19 - 23.516}{8} = \frac{-4.516}{8} = -0.5645\).
  • Step 10: Multiply roots: \(5.3145 \times (-0.5645) = -3.00...\)
  • Risk of arithmetic errors at every step!
✅ The Practix Way (1s)
  • Step 1: \(-12 / 4 = -3\)
Quadratics

4. Sum of Solutions

\[ \text{Sum} = -\frac{b}{a} \]

Find the sum of all x-values without solving.

"Sum of solutions for \(2x^2 + 10x - 48 = 0\)."

🚫 The School Way (50+ Seconds)
  • Step 1: Look at \(2x^2 + 10x - 48 = 0\). Too messy to factor directly.
  • Step 2: Divide everything by 2: \(x^2 + 5x - 24 = 0\).
  • Step 3: Find two numbers that multiply to -24 AND add to 5.
  • Step 4: Try pairs: (1,-24)? No. (2,-12)? No. (3,-8)? No...
  • Step 5: Continue testing: (4,-6)? No. (6,-4)? No...
  • Step 6: Finally find (8,-3): \(8 \times (-3) = -24\), \(8 + (-3) = 5\). ✓
  • Step 7: Write factored form: \((x + 8)(x - 3) = 0\).
  • Step 8: Solve: \(x + 8 = 0\) gives \(x_1 = -8\).
  • Step 9: Solve: \(x - 3 = 0\) gives \(x_2 = 3\).
  • Step 10: Add the roots: \(-8 + 3 = -5\).
  • Mental fatigue from all that trial-and-error!
✅ The Practix Way (1s)
  • Step 1: \(-10 / 2 = -5\)
Algebra

5. Complementary Trig

\[ \sin(x) = \cos(90 - x) \]

Sine of one angle equals cosine of its complement.

Geometry

6. Exterior Angle Sum

Always \(360^\circ\)

The exterior angles of ANY polygon always sum to 360. Instant time saver.

Quadratics

7. The Discriminant

\[ D = b^2 - 4ac \]

Determine the number of solutions instantly.

"How many real solutions for \(x^2 + 4x + 5 = 0\)?"

🚫 The School Way (40+ Seconds)
  • Step 1: Look at \(x^2 + 4x + 5 = 0\).
  • Step 2: Try to factor: Need numbers that multiply to 5, add to 4.
  • Step 3: Try (1,5): \(1 + 5 = 6\). Nope.
  • Step 4: No other integer pairs work. Factoring fails!
  • Step 5: Resort to Quadratic Formula: \(x = \frac{-4 \pm \sqrt{16-20}}{2}\).
  • Step 6: Calculate discriminant: \(4^2 - 4(1)(5) = 16 - 20 = -4\).
  • Step 7: Notice it's negative. Can't take square root of negative!
  • Step 8: Conclude: Zero real solutions.
  • Wasted time on failed factoring attempt!
✅ The Practix Way
  • Step 1: \(4^2 - 4(1)(5) = -4\).
  • Step 2: Negative = Zero real solutions.
Algebra

8. Weighted Average

\[ \text{Avg} = \frac{\sum w \cdot x}{\sum w} \]

Essential for mixture problems and grouped data.

Exclusive Hack
Algebra

9. Harmonic Mean (Avg Speed)

\[ \text{Speed} = \frac{2ab}{a+b} \]

When distances are equal, your average speed is NOT the simple average.

Geometry

10. Distance from Origin

\[ d = \sqrt{x^2 + y^2} \]

A specialized version of the distance formula for points at \((0,0)\).

Algebra

11. Growth & Decay

\[ y = a(1 \pm r)^t \]

Master percent word problems.

"$1000 grows at 5% for 3 years."

🚫 The Manual Way (45+ Seconds)
  • Step 1: Start with $1000. Increase by 5%.
  • Step 2: Calculate: \(1000 \times 0.05 = 50\).
  • Step 3: Add to original: \(1000 + 50 = 1050\).
  • Step 4: Year 2: \(1050 \times 0.05 = 52.50\).
  • Step 5: Add again: \(1050 + 52.50 = 1102.50\).
  • Step 6: Year 3: \(1102.50 \times 0.05 = 55.125\).
  • Step 7: Final addition: \(1102.50 + 55.125 = 1157.625\).
  • Step 8: Round to $1157.63.
  • Repetitive, error-prone, doesn't scale to 10+ years!
✅ The Formula Way
  • Step 1: \(1000(1.05)^3\)
Algebra

12. Percent Change

\[ \frac{\text{New} - \text{Old}}{\text{Old}} \times 100 \]

Calculate increase/decrease without confusion.

Geometry

13. Arc/Sector Proportions

\[ \frac{\text{Arc}}{\text{Circum}} = \frac{\text{Sector}}{\text{Area}} = \frac{\theta}{360} \]

The universal proportion for all circle slice problems.

Geometry

14. Mastery: The 3-4-5 & 5-12-13 Triangles

3 : 4 : 5 Patterns

Spot these Pythagoras patterns to avoid doing math entirely.

Geometry

15. Slope Formula

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Find the steepness of any line through two points.

Geometry

16. Parallel Lines

\[ m_1 = m_2 \]

Parallel lines have identical slopes.

Geometry

17. Perpendicular Lines

\[ m_1 \cdot m_2 = -1 \]

Perpendicular slopes are negative reciprocals.

Statistics

18. Standard Deviation Logic

Spread = Gap

Don't calculate it. Understand it. More "spread out" data = Higher Standard Deviation.

Statistics

19. Margin of Error

Sample Size ↑ = MoE ↓

The more people you ask, the smaller the error. Guaranteed point if you remember this inverse relationship.

Geometry

20. Sum of Interior Angles

\[ \text{Sum} = (n-2) \cdot 180^\circ \]

Find the total degrees inside any polygon with \(n\) sides.

Geometry

21. Equation of Circle (General Form)

\( x^2 + y^2 + Dx + Ey + F = 0 \)

Use "Halve and Square" to jump straight to the center and radius.