It Always Works
Part of the Factoring & Quadratics section of Coddy's Algebra journey. Lesson 51 of 63.
Here is the formula's real power. x^2 + 3x + 1 = 0: the machine finds no pair, because none exists over whole numbers. The formula does not care: x = (-3 ± √(3^2 - 4 * 1 * 1)) / 2 = (-3 ± √5) / 2.
The discriminant 5 is not a perfect square, so the root stays as the radical √5, exactly as in the square-root chapter. The two solutions are exact irrational numbers, and the formula wrote them down without complaint.
That is the division of labor: factoring is fast when it works, the formula works always. When the discriminant is a perfect square you could have factored; when it is not, the formula was the only road, and the radical in the answer is the proof.
Challenge
HardSolve x^2 + 5x + 2 = 0 exactly. The radical will survive to the answer, and that is correct. Factoring chips are off.
Try it yourself
Solve for x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Factoring & Quadratics
1What Factoring Is
Two DirectionsThe Reverse GearUn-Distributing LettersThe Check Is MultiplicationRecap: First Factors4Minus Signs in the Machine
A Negative ProductWhich One Gets the MinusBoth NegativeThe Sign MapRecap: Signs7The Zero Product Rule
A Product That Makes ZeroOne Factor Is EnoughThe Bare XEvery Factor Gets a TurnRecap: Zero Product10The Quadratic Formula
The Formula, StatedThe Number Under the RootSplit the Plus-MinusIt Always WorksChoosing a MethodRecap: The Formula2The Greatest Common Factor
Number and Letter TogetherThe Largest OneHigher PowersThree Terms Share TooRecap: The GCF5Special Shapes
A Missing MiddleThe Shape in LettersThe Perfect SquareThe Minus VersionPrime PolynomialsRecap: Special Shapes3The Sum-Product Machine
Reading the ProductHunting the PairThe Machine in LettersLonger HuntsRecap: The Machine6Factor Completely
Two LayersSquares InsideCompletely Means CompletelyThe Deep CheckRecap: Factor Completely