### Teorija

Ar Vjeta teorēmu var atrisināt kvadrātvienādojumu.
Parasti Vjeta teorēmu lieto reducētam kvadrātvienādojumam, t.i., ja koeficients $a=1$.
${x}^{2}+\mathit{px}+q=0 ⇒\phantom{\rule{0.147em}{0ex}}\left\{\begin{array}{l}{x}_{1}\cdot {x}_{2}=q\\ {x}_{1}+{x}_{2}=-p\end{array}\right\$
Piemērs:
Nosaki saknes!
$\begin{array}{l}{x}^{2}-14x+40=0\\ \left\{\begin{array}{l}{x}_{1}\cdot {x}_{2}=40\\ {x}_{1}+{x}_{2}=14\end{array}\right\\\ {x}_{1}=10\\ {x}_{2}=4\end{array}$

Arī kvadrātvienādojumam, kurā $a\ne 1$, ir spēkā Vjeta teorēma.

$\begin{array}{l}a{x}^{2}+\mathit{bx}+c=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}|:a\\ \\ \frac{a}{a}{x}^{2}+\frac{b}{a}x+\frac{c}{a}=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}⇒\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{x}^{2}+\frac{b}{a}x+\frac{c}{a}=0\\ \\ \left\{\begin{array}{l}{x}_{1}\cdot {x}_{2}=\frac{c}{a}\\ {x}_{1}+{x}_{2}=-\frac{b}{a}\end{array}\right\\end{array}$
(${x}_{1}$ un ${x}_{2}$ ir vienādojuma saknes)

Piemērs:
Nosaki saknes, izmantojot Vjeta teorēmu!
$\begin{array}{l}12{x}^{2}+x-1=0\\ \\ \frac{12}{12}{x}^{2}+\frac{1}{12}x-\frac{1}{12}=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}⇒\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{x}^{2}+\frac{1}{12}x-\frac{1}{12}=0\\ \\ \left\{\begin{array}{l}{x}_{1}\cdot {x}_{2}=-\frac{1}{12}\\ {x}_{1}+{x}_{2}=-\frac{1}{12}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.588em}{0ex}}\end{array}\right\\phantom{\rule{0.441em}{0ex}}\phantom{\rule{0.147em}{0ex}}{x}_{1\phantom{\rule{0.147em}{0ex}}}=-\frac{1}{3}\phantom{\rule{0.588em}{0ex}}{x}_{2}=\frac{1}{4}\phantom{\rule{0.147em}{0ex}}\end{array}$

Ja, izmantojot Vjeta teorēmu, ir grūti uzminēt saknes, tās var rēķināt ar citām metodēm un tad  ar Vjeta teorēmu var pārbaudīt, vai kvadrātvienādojuma saknes ir izrēķinātas pareizi.

Piemērs:
$\begin{array}{l}{x}^{2}+0,8x-0,1=0\\ D={b}^{2}-4\mathit{ac}={0,8}^{2}-4\cdot 2\cdot \left(-0,1\right)=1,44\\ {x}_{1}=\frac{-b+\sqrt{D}}{2a}=\frac{-0,8+1,2}{2\cdot 2}=0,1\\ {x}_{2}=\frac{-b-\sqrt{D}}{2a}=\frac{-0,8-1,2}{2\cdot 2}=-0,5\end{array}$

Pārbaude:
$\begin{array}{l}2{x}^{2}+0,8x-0,1=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}|:2 \\ {x}^{2}+0,4x-0,05=0\\ \\ \left\{\begin{array}{l}0,1\cdot \left(-0,5\right)=-0,05\\ 0,1-0,5=-0,4\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\end{array}\right\\end{array}$

Ja pilna sakņu pārbaude šķiet sarežģīta, tad vismaz vajag pārbaudīt sakņu zīmju pareizību. Šajā piemērā redzams, ka saknēm ir jābūt ar atšķirīgām zīmēm, jo $c<0$.

Izmantojot Vjeta teorēmu, var sastādīt kvadrātvienādojumu, ja ir zināmas tā saknes.
Piemērs:
Kāda kvadrātvienādojuma saknes ir $$2$$ un $$-0,3$$?

$\begin{array}{l}{x}^{2}+\mathit{px}+q=0\\ 2+\left(-0,3\right)=1,7\phantom{\rule{0.147em}{0ex}}=-p \phantom{\rule{0.147em}{0ex}}⇒\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}p=-1,7\\ 2\cdot \left(-0,3\right)=-0,6=q\\ \\ {x}^{2}-1,7x-0,6=0\end{array}$

* Fransuā Vjets (1540 -1603) ir franču matemātiķis. Pēc izglītības - jurists.