Core Math Formulas Everyone Should Know (With Real Examples)

A small set of formulas covers the vast majority of math questions people actually search for. Here's what's happening under the hood of the most commonly used ones.

Percentages: three questions, one formula

Almost every percentage question is a variation of "part over whole times 100." What is 25 as a percentage of 200? (25/200) × 100 = 12.5%. What's 18% of 50? 50 × 0.18 = 9. What's the percent change from 50 to 75? ((75−50)/50) × 100 = 50%.

The only thing that changes between these three "types" of percentage question is which number is unknown — the math itself never changes.

The quadratic formula, demystified

Any equation of the form ax² + bx + c = 0 can be solved with:

x = (−b ± √(b² − 4ac)) / 2a

Take x² − 3x + 2 = 0: here a=1, b=−3, c=2. The discriminant (b² − 4ac) is 9 − 8 = 1, so √1 = 1, giving x = (3 ± 1) / 2 — two roots, x = 2 and x = 1. You can check this by factoring: (x−1)(x−2) = 0, which confirms both roots.

If the discriminant comes out negative, there's no real solution — the parabola never crosses the x-axis, only complex roots exist, which is a separate topic entirely.

The Pythagorean theorem in the real world

a² + b² = c² isn't just a geometry class exercise — it's how you calculate diagonal distances, screen sizes, and cable lengths. A ladder leaning against a wall with its base 3 meters out and reaching 4 meters up the wall needs a length of √(3² + 4²) = √25 = 5 meters. That's the famous 3-4-5 triangle, and it works for any right triangle, not just that specific ratio.

Sequences: arithmetic vs. geometric

An arithmetic sequence adds the same amount each step (3, 8, 13, 18, ...). A geometric sequence multiplies by the same ratio each step (2, 6, 18, 54, ...). The formulas differ because the underlying growth is different: arithmetic sequences grow linearly, geometric sequences grow exponentially — the same distinction as simple vs. compound interest, just abstracted away from money.

Try it yourself

• The percentage calculator and percentage change calculator cover all three percentage question types above
• The quadratic equation solver shows the discriminant explicitly so you can see why some equations have no real solution
• The Pythagorean theorem calculator and distance formula calculator both solve variations of the same underlying right-triangle relationship