Math Principles & Formulas — Every Essential Rule Explained

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Algebra

Quadratic Formula

\[x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\]

Solves any equation of the form \(ax^{2}+bx+c=0\). The expression under the root (the discriminant) tells you how many real solutions exist.

Example: \(x^{2}-5x+6=0\) gives \(x=2\) and \(x=3\).

Distributive Property

\[a(b+c)=ab+ac\]

Multiplying a sum by a number is the same as multiplying each term separately and adding the results. The foundation of expanding brackets.

Example: \(3(x+4)=3x+12\)

Difference of Squares

\[a^{2}-b^{2}=(a+b)(a-b)\]

Any difference of two perfect squares factors instantly into a sum times a difference — one of the most used factoring shortcuts.

Example: \(x^{2}-9=(x+3)(x-3)\)

Exponent Rules

\[a^{m}\cdot a^{n}=a^{m+n}\qquad \frac{a^{m}}{a^{n}}=a^{m-n}\qquad (a^{m})^{n}=a^{mn}\]

Multiplying powers with the same base adds exponents; dividing subtracts them; a power of a power multiplies them.

Example: \(2^{3}\cdot 2^{4}=2^{7}=128\)

Logarithm Rules

\[\log(ab)=\log a+\log b\qquad \log\frac{a}{b}=\log a-\log b\qquad \log a^{n}=n\log a\]

Logarithms turn multiplication into addition and powers into multiplication — the reason they simplify huge calculations.

Example: \(\log_{10}1000=3\) because \(10^{3}=1000\).

Cross-Multiplication

\[\frac{a}{b}=\frac{c}{d}\;\Rightarrow\; a\cdot d=b\cdot c\]

When two fractions are equal, the diagonal products are equal. The fastest way to solve proportions.

Example: \(\frac{9}{27}=\frac{x}{45}\Rightarrow x=15\)

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Geometry

Pythagorean Theorem

\[a^{2}+b^{2}=c^{2}\]

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. Over 2,500 years old and still everywhere.

Example: legs 3 and 4 → hypotenuse \(\sqrt{9+16}=5\).

Circle: Area & Circumference

\[A=\pi r^{2}\qquad C=2\pi r\]

Both depend only on the radius. Doubling the radius doubles the circumference but quadruples the area.

Example: \(r=5\Rightarrow A=25\pi\approx 78.5\)

Triangle Area

\[A=\frac{1}{2}\,b\,h\]

Half the base times the height — because every triangle is exactly half of a parallelogram with the same base and height.

Example: \(b=10,\;h=6\Rightarrow A=30\)

Sum of Interior Angles

\[S=(n-2)\times 180^{\circ}\]

Any polygon with \(n\) sides splits into \(n-2\) triangles, each contributing 180°. A triangle: 180°. A square: 360°.

Example: hexagon → \((6-2)\times180=720^{\circ}\)

Volume of Cylinder & Sphere

\[V_{cyl}=\pi r^{2}h\qquad V_{sph}=\frac{4}{3}\pi r^{3}\]

A cylinder is a circle stretched to height \(h\); a sphere's volume is exactly two-thirds of the cylinder that contains it — Archimedes' favorite result.

Example: \(r=3,\;h=10\Rightarrow V_{cyl}=90\pi\)

Distance Formula

\[d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\]

The distance between two points is just the Pythagorean theorem applied to the coordinate grid.

Example: from \((0,0)\) to \((3,4)\): \(d=5\)

Trigonometry

Pythagorean Identity

\[\sin^{2}\theta+\cos^{2}\theta=1\]

True for every angle, because \((\cos\theta,\sin\theta)\) is always a point on the unit circle. The most important identity in trig.

Example: if \(\sin\theta=0.6\) then \(\cos\theta=\pm0.8\).

Law of Sines

\[\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\]

In any triangle, each side divided by the sine of its opposite angle gives the same ratio. Solves triangles that aren't right-angled.

Law of Cosines

\[c^{2}=a^{2}+b^{2}-2ab\cos C\]

The Pythagorean theorem with a correction term for the angle. When \(C=90^{\circ}\), the cosine vanishes and it becomes \(a^2+b^2=c^2\).

SOH-CAH-TOA

\[\sin\theta=\frac{opp}{hyp}\qquad \cos\theta=\frac{adj}{hyp}\qquad \tan\theta=\frac{opp}{adj}\]

The three basic ratios in a right triangle — the entry point to all of trigonometry.

Example: \(\sin 30^{\circ}=\frac{1}{2}\), \(\cos 60^{\circ}=\frac{1}{2}\)

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Calculus

Power Rule

\[\frac{d}{dx}x^{n}=n\,x^{n-1}\]

To differentiate a power of \(x\): bring the exponent down as a multiplier, then reduce the exponent by one.

Example: \(\frac{d}{dx}x^{3}=3x^{2}\)

Product & Quotient Rules

\[(uv)'=u'v+uv'\qquad \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^{2}}\]

Derivatives don't distribute over multiplication — these rules handle products and fractions of functions correctly.

Chain Rule

\[\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)\]

For a function inside a function: differentiate the outside, keep the inside, then multiply by the derivative of the inside.

Example: \(\frac{d}{dx}\sin(x^{2})=2x\cos(x^{2})\)

Fundamental Theorem of Calculus

\[\int_{a}^{b}f(x)\,dx=F(b)-F(a)\]

Integration and differentiation are inverse operations: the area under a curve equals the change in its antiderivative. This links the two halves of calculus.

Example: \(\int_{0}^{\pi}\sin x\,dx=2\)

Physics

Newton's Second Law

\[F=ma\]

Force equals mass times acceleration. The heavier the object, the more force you need to accelerate it the same amount.

Example: \(2\,\text{kg}\) at \(3\,\text{m/s}^2\) needs \(6\,\text{N}\).

Mass–Energy Equivalence

\[E=mc^{2}\]

Mass is a concentrated form of energy. Because \(c^{2}\) is enormous, a tiny amount of mass holds a staggering amount of energy.

Ohm's Law

\[V=IR\]

Voltage equals current times resistance — the single most used relation in electric circuits.

Example: \(2\,\text{A}\) through \(5\,\Omega\) → \(10\,\text{V}\)

Kinetic Energy

\[KE=\frac{1}{2}mv^{2}\]

Energy of motion grows with the square of speed: doubling your speed quadruples your kinetic energy — the physics behind braking distances.

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Space & Astrophysics

Newton's Law of Universal Gravitation

\[F=\frac{G\,M\,m}{r^{2}}\]

Every two masses attract each other, weakening with the square of the distance. It governs apples and moons alike.

Example: Earth–Moon force ≈ \(1.98\times10^{20}\,\text{N}\)

Escape Velocity

\[v_{e}=\sqrt{\frac{2GM}{r}}\]

The minimum launch speed needed to break free of a planet's gravity — independent of the object's own mass.

Example: Earth: \(v_{e}\approx 11.2\,\text{km/s}\)

Kepler's Third Law

\[T^{2}=\frac{4\pi^{2}}{GM}\,a^{3}\]

A planet's orbital period squared is proportional to its distance cubed. Farther planets don't just travel more — they travel slower.

Example: Mars at 1.524 AU → \(T\approx1.88\) years

Quantum Physics

Planck's Photon Energy

\[E=hf=\frac{hc}{\lambda}\]

Light comes in packets whose energy depends only on frequency. Higher frequency (bluer) light carries more energy per photon.

Example: a 500 nm photon carries ≈ 2.48 eV.

de Broglie Wavelength

\[\lambda=\frac{h}{p}\]

Every moving particle behaves like a wave. The faster and heavier it is, the shorter its wavelength — why we don't notice our own.

Heisenberg Uncertainty Principle

\[\Delta x\cdot\Delta p\geq\frac{\hbar}{2}\]

Position and momentum cannot both be known precisely at once — not a measurement flaw, but a fundamental property of nature.

Bohr Model Energy Levels

\[E_{n}=-\frac{13.6}{n^{2}}\;\text{eV}\]

Electrons in hydrogen occupy fixed energy levels. Jumping between levels emits or absorbs photons of exact energies — the origin of atomic spectra.

Example: \(n=2\Rightarrow E_{2}=-3.4\,\text{eV}\)

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