Solve x² − 2x + 1 = 0

Every quadratic equation can be arranged as ax² + bx + c = 0. Reading off a = 1, b = -2 and c = 1, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.

The goal is to write x² + bx + c as (x + p)(x + q). Multiplying that out gives x² + (p + q)x + p·q, so p and q must multiply to c = 1 and add up to b = -2.

List the pairs of numbers whose product is 1, with their signs, and pick the pair whose sum is -2.

0 is the only number with this property: if a·b = 0, then a = 0 or b = 0. That is why the equation was first rearranged to … = 0 and factored: now each factor can be set to 0 on its own, giving a simpler equation for each.

An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.

A factor that appears more than once, like (x − 2)², gives the same root again. The root is counted that many times; when the count is even, the graph touches the x-axis there instead of crossing it.