Solving by Factoring & the Square-Root Property

The zero-product property

Here is the one fact that makes factoring work: if a product of two things equals zero, then at least one of them is zero. That is the [[zero-product-property|zero-product property]], and nothing but zero on the right-hand side will do — this is exactly why we insist on standard form first. Once a quadratic is written as (something)·(something) = 0, each factor gives its own simple equation.

1. Write the quadratic in standard form so one side is 0.
2. Factor the left side into a product of two linear factors.
3. Set each factor equal to 0 and solve the small linear equations.
4. Each root is a value where the curve crosses the x-axis — check it by substitution.

Solve  x^2 - 5x + 6 = 0  by factoring.

  Need two numbers multiplying to +6, adding to -5:
  those are -2 and -3.

    x^2 - 5x + 6 = 0
    (x - 2)(x - 3) = 0

  Zero-product property:
    x - 2 = 0   or   x - 3 = 0
    x = 2       or   x = 3

  Check x = 2:  4 - 10 + 6 = 0  ✓
  Check x = 3:  9 - 15 + 6 = 0  ✓

When there is no middle term: the square-root property

If a quadratic has no x term — only an x^2 and a constant — you do not need to factor at all. Isolate the square and take the square root of both sides. This is the [[square-root-property|square-root property]], and its crucial detail is the plus-or-minus: if x^2 = 9, then x could be +3 or -3, because both square to 9. Forgetting the negative root is the single most common mistake here.

Square-root property:  if  u^2 = k (k ≥ 0),  then  u = ±sqrt(k).

Solve  2(x - 1)^2 - 18 = 0:

    2(x - 1)^2 = 18
      (x - 1)^2 = 9
       x - 1 = ±3        (take the square root, keep ±)
       x - 1 = 3   or   x - 1 = -3
       x = 4       or   x = -2