Ch 1.4.9 – Of the Nature of Equations of the Second Degree
Summary: Explains why quadratics always have exactly two solutions by factoring into linear factors, derives Vieta’s formulas relating roots to coefficients, determines when roots are real vs. imaginary via the discriminant, and shows how a known root yields the second by polynomial division.
Sources: chapter-1.4.9
Last updated: 2026-05-03
The fundamental principle
Any product of factors equals zero if and only if at least one factor is zero (source: chapter-1.4.9, §692). This is the key that unlocks the double solution.
Example: factors as . The two solutions and arise because each factor can be set to zero independently.
Every quadratic has exactly two solutions
A degree-2 polynomial always factors into exactly two linear factors . Three factors would give degree 3; one factor degree 1. Therefore every quadratic has exactly two roots — no more, no less (§703). The roots may be:
- Two distinct real values
- A repeated real value ()
- Two complex conjugate values
Vieta’s formulas (§695)
Expanding and comparing with :
These hold even when the roots are imaginary. For example, has roots and ; their sum is and their product is (§701).
See vieta-formulas.
Sign rules for roots (§696–697)
For the equation :
| Signs of term and | Nature of roots |
|---|---|
| , (i.e. ) | Both roots positive |
| , (i.e. ) | One positive, one negative |
| , (i.e. ) | Both roots negative |
Constructing a quadratic from given roots (§698)
To form an equation with roots and : write . Example: roots and give .
Equal roots (§699)
When both roots are equal (), the quadratic is a perfect square: . Solving gives twice.
The discriminant — real vs. imaginary roots (§700–702)
For , the solution is .
- If : two distinct real roots.
- If : two equal real roots.
- If (equivalently ): both roots are imaginary — the equation is impossible.
For the general form , the roots are imaginary when , i.e., four times the product of the leading and constant coefficients exceeds the square of the linear coefficient (§702). See discriminant.
Key remark (§702): imaginary roots admit no approximation (unlike irrational real roots), since no real number can be made arbitrarily close to .
Finding the second root from the first (§704–705)
If is one root of the quadratic , then is a factor. Polynomial long division gives the other linear factor , and hence the second root.
Example: knowing is a root of , divide by to obtain , giving the second root .