What is the difference between linear and quadratic equations?

Linear equations contain x to the first power only and have at most one solution. Quadratics contain x² and have up to two solutions, found via factoring or the quadratic formula.

Do I need to memorize the quadratic formula?

Yes — it is one of the highest-value memorizations in school math. It solves every quadratic, including ones with irrational or complex roots that factoring cannot touch.

Why do I keep getting sign errors?

Almost always from moving terms across "=" without flipping the sign, or from distributing a minus over parentheses. Slow down on exactly those two moves and re-check them.

Algebra, start to finish

Everything in the algebra section, explained in one place: linear equations, quadratics, and slope, then practice with the calculators.

Algebra is the moment math stops being about numbers and starts being about structure. Nearly every algebra problem is one of three shapes: a linear equation (find x), a quadratic (x² involved), or a slope question (how fast things change). Learn to recognize the shape and you already know the method.

1. Linear equations

A linear equation has x to the first power only, no x², no √x. The entire game is isolating x: do the same operation to both sides until x stands alone.

ax + b = cx + d → x = (d − b) / (a − c)

Worked example: 5x − 7 = 3x + 9.

Move x-terms left, constants right: 5x − 3x = 9 + 7 → 2x = 16.
Divide: x = 8.
Check (never skip): left = 5(8) − 7 = 33; right = 3(8) + 9 = 33. ✓

Two special cases: if you reach 0 = 0, every x works (the equation is an identity, e.g. 2x + 4 = 2(x + 2)). If you reach 0 = 5, nothing works, the equation contradicts itself.

Solve linear equations with steps →

2. Quadratic equations

Anything with x² is quadratic. First rearrange to standard form ax² + bx + c = 0, then the quadratic formula finishes it, always, even when factoring fails.

x = (−b ± √(b² − 4ac)) / 2a

Worked example: x² − 5x + 6 = 0. Here a = 1, b = −5, c = 6.

Discriminant: b² − 4ac = 25 − 24 = 1.
x = (5 ± √1) / 2 = (5 ± 1) / 2, so x = 3 or x = 2.
Check x = 3: 9 − 15 + 6 = 0 ✓

Read the discriminant first

b² − 4ac predicts the answer type: positive → two real roots; zero → one repeated root; negative → two complex roots (the parabola never touches the x-axis).

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3. Slope

Slope measures steepness as a rate: for every 1 unit you move right, how many units do you move up?

m = (y₂ − y₁) / (x₂ − x₁)

Worked example: through (2, 3) and (6, 11): m = (11 − 3)/(6 − 2) = 8/4 = 2. The line climbs 2 units for every 1 unit right.

Watch the edge cases: vertical line → division by zero → slope is undefined (not zero). Horizontal line → rise is 0 → slope is 0.

Compute slope with steps →

The three mistakes that cost the most points

Sign errors when moving terms: 5x = 3x + 9 becomes 2x = 9, not 8x = 9. Moving across “=” flips the sign, every time, no exceptions.
Forgetting the ± : x² = 16 gives x = 4 or x = −4. The square root of a positive number has two answers.
Dividing by zero disguised: in slope, x₂ = x₁ means the points stack vertically, stop and say “undefined” instead of computing.

Keep practicing

Type any equation into the universal solver, it detects linear vs quadratic automatically and shows every step. Or browse all algebra calculators.