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Lesson 4 · The quadratic formula
The formula that never misses
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The formula that never misses

When factoring stalls, the formula does not: x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Read aloud: x equals minus b, plus or minus the square root of b-squared minus four a c, all over two a — a formula you can say is a formula you own. The ± is two answers in one line: a plus branch and a minus branch, one for each root.

With a positive, plus digs out the LARGER root and minus the smaller — the branches never trade places. Two disciplines settle almost every mark: the whole crown, −b and square root together, sits over 2a — all of it; and b walks in wearing its own sign, so for x2−6x+8x^2 - 6x + 8 the formula opens with −b=+6-b = +6. Most formula wounds are sign wounds.

x

x=−b+b2−4ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

  • xx= Root (plus branch)
  • aa= Coefficient of x²
  • bb= Coefficient of x
  • cc= Constant term
Quadratic Formula (Positive Root) solver →
x

x=−b−b2−4ac2ax = \frac{-b - \sqrt{b^2 - 4ac}}{2a}

  • xx= Root (minus branch)
  • aa= Coefficient of x²
  • bb= Coefficient of x
  • cc= Constant term
Quadratic Formula (Negative Root) solver →