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Perfectio quadri est ars algebrae elementariae in qua possumus reponere hanc expressionem

x2+bxdisplaystyle x^2+bx

cum

(x+c)2+ddisplaystyle (x+c)^2+d

Presse habemus:

ax2+bx+c=a(x2+bxa)+c=a(x2+bxa+(b24a2−b24a2))+c=a(x2+2bx2a+(b2a)2)−b24a+c=a(x+b2a)2−b24a+cdisplaystyle beginmatrixax^2+bx+c&=&aleft(x^2+frac bxaright)+c\&=&aleft(x^2+frac bxa+left(frac b^24a^2-frac b^24a^2right)right)+c\&=&aleft(x^2+2frac bx2a+left(frac b2aright)^2right)-frac b^24a+c\&=&aleft(x+frac b2aright)^2-frac b^24a+cendmatrix

Quadro perfacto, ulla formula cum quadratico polynomiale reduci ad unam cum quadratico polynomiale quadrato et constante potest.

Exemplum |

• Exemplum facile est:

x2+4x=x2+4x+4−4=(x+2)2−4displaystyle x^2+4x=x^2+4x+4-4=(x+2)^2-4

• Nunc, difficilius exemplum adest in hoc antiderivativum inveniendo:

∫dx9x2−90x+241.displaystyle int frac dx9x^2-90x+241.

Denominator est

9x2−90x+241=9(x2−10x)+241.displaystyle 9x^2-90x+241=9(x^2-10x)+241.

Addendo (10/2)² = 25 to x² - 10x dat quadrum perfectum x² - 10x + 25 = (x - 5)². Ita invenimus

9(x2−10x)+241=9(x2−10x+25)+241−9(25)=9(x−5)2+16.displaystyle 9(x^2-10x)+241=9(x^2-10x+25)+241-9(25)=9(x-5)^2+16.

Sine integrale nostrum esse:

∫dx9x2−90x+241=19∫dx(x−5)2+(4/3)2=19⋅34arctan⁡3(x−5)4+C.displaystyle int frac dx9x^2-90x+241=frac 19int frac dx(x-5)^2+(4/3)^2=frac 19cdot frac 34arctan frac 3(x-5)4+C.

Nexus interni

• Aequatio quadratica


• Functio quadratica

Nexus externus |

• Perfectio quadri apud PlanetamMathematicam