Finding a formula from an equation

The idea here is pattern recognition: an unfamiliar-looking equation is often a standard formula wearing different letters. Once you see which formula it matches, you know what each part means and how to work with it.

Geometry formulas are common. Anything of the form $\frac{4}{3}\pi(\ldots)^3$ is a sphere volume with $(\ldots)$ as the radius, and $\pi r^2 h$ is a cylinder volume. Matching the equation to the formula tells you which quantity is the radius, the height, and so on.

Some equations are hidden binomial products. $x^2 - y^2$ is a difference of squares, so $x^2 - y^2 = (x - y)(x + y)$, and $x^2 + 6x + 9$ is a perfect square, $(x + 3)^2$. Spotting the pattern lets you factor or expand in one step instead of grinding through algebra.

Relationships also come as familiar function forms. $F = \frac{9}{5}C + 32$ is a linear function $y = mx + b$ with slope $\frac{9}{5}$ and intercept $32$; $P = 250(0.985)^t$ is an exponential $a \cdot b^t$ with starting value $250$ and factor $0.985$. Recognizing the form hands you the rate, the starting value, and how the quantity behaves.

A tank's volume is $V = \pi(4 + r)^2 h$. What formula is this, and what is the radius?
It matches the cylinder-volume formula $\pi R^2 h$, so it is a cylinder with radius $R = 4 + r$.
At $r = 0$ the original radius is $4$.

Rewrite $x^2 - 16$ as a product.
It is a difference of squares, $x^2 - 4^2$, which matches $a^2 - b^2 = (a - b)(a + b)$.
So $x^2 - 16 = (x - 4)(x + 4)$.

A population model is $P = 800(1.05)^t$. What is the starting value and the yearly growth?
It matches the exponential form $a \cdot b^t$, so the starting value is $a = 800$ and the factor is $b = 1.05$ — a $5\%$ increase each year.