et l’on aurait
![{\displaystyle {\begin{aligned}&{\sqrt {\alpha +\beta x+\gamma x^{2}+\delta x^{3}+\varepsilon x^{4}}}-{\sqrt {\alpha +\beta y+\gamma y^{2}+\delta y^{3}+\varepsilon y^{4}}}\\&\qquad =(x-y){\sqrt {\mathrm {G} ^{2}+\delta (x+y)+\varepsilon (x+y)^{2}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd622524190d92703ca564ba9923f307efe58e1d)
14. La différence des deux quantités qui sont sous le signe étant
![{\displaystyle \beta (x-y)+\gamma \left(x^{2}-y^{2}\right)+\delta \left(x^{3}-y^{3}\right)+\varepsilon \left(x^{4}-y^{4}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f6451505dc958a29dd7a024b9c59f3be6475e96)
on aura par l’équation (D)
![{\displaystyle {\sqrt {\alpha +\beta x+\gamma x^{2}+\delta x^{3}+\varepsilon x^{4}}}-{\sqrt {\alpha +\beta y+\gamma y^{2}+\delta y^{3}+\varepsilon y^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b40f25c6c7b92f89c4d63d07ece311a87ed48a5)
![{\displaystyle {\begin{aligned}&={\frac {\beta (x-y)+\gamma \left(x^{2}-y^{2}\right)+\delta \left(x^{3}-y^{3}\right)+\varepsilon \left(x^{4}-y^{4}\right)}{(x-y){\sqrt {\mathrm {G} ^{2}+\delta (x+y)+\varepsilon (x+y)^{2}}}}}\\&={\frac {\beta +\gamma (x+y)+\delta \left(x^{2}+xy+y^{2}\right)+\varepsilon \left(x^{3}+x^{2}y+xy^{2}+y^{3}\right)}{\sqrt {\mathrm {G} ^{2}+\delta (x+y)+\varepsilon (x+y)^{2}}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b3c3380094e23d61cb8dcdaf4d4e8d5cb908211)
donc, combinant cette équation avec celle que nous venons de citer, on aura
![{\displaystyle {\sqrt {\alpha +\beta x+\gamma x^{2}+\delta x^{3}+\varepsilon x^{4}}}={\frac {(x-y){\sqrt {\mathrm {G} ^{2}+\delta (x+y)+\varepsilon (x+y)^{2}}}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2de62915036450957089ab823202049158c83786)
![{\displaystyle +{\frac {\beta +\gamma (x+y)+\delta \left(x^{2}+xy+y^{2}\right)+\varepsilon \left(x^{3}+x^{2}y+xy^{2}+y^{3}\right)}{2{\sqrt {\mathrm {G} ^{2}+\delta (x+y)+\varepsilon (x+y)^{2}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e43f1e9b06d8fdbbb3327d53656d2c79c41a5b90)
![{\displaystyle ={\frac {\beta +\gamma (x+y)+\mathrm {G} (x-y)+\delta \left(2x^{2}+xy\right)+2\varepsilon \left(x^{3}+x^{2}y\right)}{2{\sqrt {\mathrm {G} ^{2}+\delta (x+y)+\varepsilon (x+y)^{2}}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2de379630a3a2ab2f9ea7750c6acd31fd2029d7a)
d’où, en multipliant en croix et quarrant les deux membres de cette équation, il viendra
![{\displaystyle {\begin{aligned}\beta ^{2}&-4\alpha \mathrm {G} +\left(2\beta \gamma +4\alpha \delta -2\beta \mathrm {G} \right)(x+y)+\left(\gamma ^{2}-4\alpha \varepsilon \right)(x+y)^{2}\\&+\mathrm {G} ^{2}(x-y)^{2}-2\beta \delta xy-2\gamma \mathrm {G} \left(x^{2}+y^{2}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16114e01aaadcd9cb76a0a3cda91f1a6b7540f32)
![{\displaystyle +\left(2\gamma \delta -4\beta \varepsilon -2\delta \mathrm {G} \right)(x+y)xy+\left(\delta ^{2}-4\varepsilon \mathrm {G} \right)x^{2}y^{2}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2d7bbbe870f86a89414db448e40c8aa1dd3b2e)
ou bien
![{\displaystyle {\begin{aligned}\beta ^{2}&-4\alpha \mathrm {G} +\left(2\beta \gamma +4\alpha \delta -2\beta \mathrm {G} \right)(x+y)\\&+\left(\gamma ^{2}-4\alpha \varepsilon \mathrm {G+G^{2}} \right)\left(x^{2}+y^{2}\right)+2\left(\gamma ^{2}-4\alpha \varepsilon -\beta \delta -\mathrm {G} ^{2}\right)xy\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2bf7045aa7171a72163a673165dd41e380fc47)
![{\displaystyle +2\left(\gamma \delta -2\beta \varepsilon -\delta \mathrm {G} \right)(x+y)xy+\left(\delta ^{2}-4\varepsilon \mathrm {G} \right)x^{2}y^{2}=0\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ca2b070cfbdb153d71ee7e7fea6136e1204f43)