vieta-formulas

Vieta’s Formulas

Summary: For a quadratic $x^2-ax+b=0$ with roots $p$ and $q$, Euler shows that $p+q=a$ (sum equals coefficient of $x$) and $pq=b$ (product equals constant term); these relations hold even when the roots are imaginary.

Sources: chapter-1.4.9, chapter-1.4.11, chapter-1.4.13

Last updated: 2026-05-03

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Derivation

Every quadratic with roots $p$ and $q$ can be written as (source: chapter-1.4.9, §694–695):

$(x-p)(x-q)=x^2-(p+q)x+pq=0$

Comparing with the standard form $x^2-ax+b=0$:

$\boxed{p+q=a}\boxed{pq=b}$

The sum of the roots equals the coefficient of $x$ (with its sign reversed), and the product of the roots equals the constant term.

Immediate consequences

• If one root is known, the other is immediately $a-p$ (from the sum) or $b/p$ (from the product).
• The sign pattern of the equation reveals the sign pattern of the roots (§697):

Equation form	Root signs
$x^2-ax+b=0$ ($a,b>0$)	Both positive
$x^2+ax-b=0$ ($a,b>0$)	One positive, one negative
$x^2+ax+b=0$ ($a,b>0$)	Both negative

Persistence for imaginary roots

The formulas hold even when $\Delta <0$ and the roots are complex. For example, $x^2-6x+10=0$ has roots $3\pm \sqrt{-1}$; their sum is $6$ and their product is $(3{)}^2-(-1)=10$, matching the coefficients (§701). See discriminant and imaginary-numbers.

Constructing equations from roots

To build a quadratic with given roots $p$ and $q$, write $(x-p)(x-q)=0$, expanding to $x^2-(p+q)x+pq=0$ (§698).

Extension to cubics

For a monic cubic $x^3-a{x}^2+bx-c=0$ with roots $p,q,r$ (source: chapter-1.4.11, §722):

$p+q+r=a,pq+pr+qr=b,pqr=c$

The constant term $c=pqr$ immediately implies the rational-root-theorem: any rational root must divide $c$.

Extension to quartics (§754–755)

For a monic quartic $x^4+a{x}^3+b{x}^2+cx+d=0$ with roots $p,q,r,s$ (source: chapter-1.4.13, §754–755):

$p+q+r+s=-a$ $pq+pr+ps+qr+qs+rs=b$ $pqr+pqs+prs+qrs=-c$ $pqrs=d$

The constant term is the product of all four roots, which immediately implies the rational-root-theorem for quartics.

Related pages

• quadratic-equations
• ch1.4.9-nature-quadratic-equations
• ch1.4.11-complete-cubic-equations
• cubic-equations
• rational-root-theorem
• discriminant
• imaginary-numbers
• completing-the-square
• quartic-equations
• ch1.4.13-quartic-equations
• equations