Algebraic Formulae

Multiplying the binomial $(a+b)$ by itself using the distributive law:

$(a+b)^2 = (a+b)(a+b) = a(a+b) + b(a+b) = a^2 + ab + ba + b^2$

Since multiplication is commutative ($ab = ba$), the like terms combine to $2ab$: $(a+b)^2 = a^2 + 2ab + b^2$.

A large square of side $(a+b)$ has total area $(a+b)^2$. Subdividing the sides into segments $a$ and $b$ divides the square into four distinct geometric regions:

• One blue square of side $a$ with area $= a^2$
• Two orange rectangles of dimensions $a \times b$ with combined area $= 2ab$
• One green square of side $b$ with area $= b^2$

Never omit coefficients: $(5m)^2 \ne 5m^2$. You must square both $5$ and $m$ to get $25m^2$.

Architecture & Land Planning: When a square plot is expanded by $a$ meters on one side and $b$ meters on the other, civil engineers calculate the additional concrete and paving needed by expanding $(a+b)^2$.

$(a-b)^2 = (a-b)(a-b) = a^2 - 2ab + b^2$.

Trinomial Square: $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.

Notice that $(-b)^2 = +b^2$, making the squared terms unconditionally positive, whereas cross-products containing one negative sign become negative ($-2ab$).

Do not write $(-y)^2 = -y^2$. The square of any real term is strictly positive ($+y^2$).

Data Science & Variance Calculation: In statistics, the squared deviation from the mean $(x_i - \bar{x})^2 = x_i^2 - 2x_i\bar{x} + \bar{x}^2$ forms the foundation of variance and standard deviation across millions of data points.

$(a+b)(a-b) = a(a-b) + b(a-b) = a^2 - ab + ba - b^2 = a^2 - b^2$.

Conversely, the difference between two perfect squares factors into the sum and difference of their bases: $a^2 - b^2 = (a+b)(a-b)$.

When two numbers have an easy midpoint, this formula delivers lightning-fast multiplication: $62 \times 58 = (60+2)(60-2) = 3600 - 4 = 3596$.

Always factor out the greatest common divisor first: in $2x^2 - 32$, factor out $2$ to get $2(x^2 - 16) = 2(x+4)(x-4)$.

Annular Ring & Hollow Pipe Cross-Sections: The cross-sectional area of a hollow metal pipe of outer radius $R$ and inner radius $r$ is $\pi R^2 - \pi r^2 = \pi(R^2 - r^2) = \pi(R+r)(R-r)$. Mechanical engineers use this formula to calculate weight and manufacturing costs.

$(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$

$(a+b)^2 - (a-b)^2 = 4ab$

$ab = \left(\frac{a+b}{2}\right)^2 - \left(\frac{a-b}{2}\right)^2$.

Whenever $(a+b)$ and $(a-b)$ are provided, $ab$ and $a^2+b^2$ can be directly computed without solving for $a$ and $b$ individually.

Do not forget to square the denominator 2: $(\frac{a+b}{2})^2 = \frac{(a+b)^2}{4}$.

Cryptography & Cybersecurity: Expressing composite odd integers as the difference of two squares is the core mechanism of Fermat's factorization algorithm, used to audit the strength of RSA cryptographic keys.

Because $x \cdot \frac{1}{x} = 1$, the cross-product simplifies to a pure constant $2$:

$x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 = (x - \frac{1}{x})^2 + 2$.

Square once to determine $x^2 + \frac{1}{x^2}$. Then square that result once more to obtain $x^4 + \frac{1}{x^4}$ systematically.

Watch transposition signs: for $(m - 1/m)^2$, the $-2$ becomes $+2$ when moved to the other side ($16 + 2 = 18$).

Electronics & Resonance Circuits: In RLC circuits, minimizing total impedance involves expressions with frequency terms $f + 1/f$. The minimum energy dissipation is derived using this reciprocal identity.