The formula
A·v + B = 0 → v = −B/A · A·v² + B·v + C = 0 → v = (−B ± √(B² − 4AC)) / 2A
Why it works
Solving for a variable is undoing, in reverse order, everything that is done to it. The calculator makes it systematic: it moves the whole formula to one side of the equals sign, expands, and groups the terms by the exponent the variable carries. If it appears with a single exponent — multiplying, which is exponent 1; dividing, which is −1; squared, which is 2; under a root, which is one half — the equation is A·vᵏ + B = 0, with A and B made of the other letters, so vᵏ = −B/A and raising to 1/k finishes it. If it appears with exponents 1 and 2 at once it is a quadratic in that variable and the general formula applies, with the other letters inside. What fits neither case — the variable inside a sine, or added in a denominator — the calculator does not solve, and says so.
How to solve it by hand
- Move everything to the same side of the equals sign by subtracting the right-hand side
- Expand and group the terms by the exponent of the variable
- If the variable has a single exponent k: vᵏ = −B/A, then raise to 1/k
- If it appears with exponents 1 and 2, apply the quadratic formula with the other letters inside
What is worth knowing
Capitals matter: R1 and R2 are different variables, and T is not t, which is what it takes to type a physical formula as it stands. E is a variable and not the number e, and π is kept as a symbol: solving a pendulum for its length gives T²g/(4π²), not 0.0253·T²g. When the variable is dividing, solving for it means flipping: from v = d/t comes t = d/v. And when it is squared, remember the two signs, which in a physical formula usually settle themselves because a negative speed makes no sense, but in an arbitrary equation are two genuine solutions. What the calculator deliberately does not do is solve from inside a function: y = sin(x) needs an arcsine and a discussion of which interval it holds on, which is no longer algebra.