frac{sqrt{2}}{2}2+1/sqrt{3}=3x^2+15/sqrt{3} eching

$$\frac{ \sqrt{ 2 } }{ 2 } 2+ \frac{ 1 }{ \sqrt{ 3 } } = \frac{ 3 { x }^{ 2 } +15 }{ \sqrt{ 3 } }$$

x uchun yechish (complex solution)

$$x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}\approx -0-1,962185028i$$

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3}\approx 1,962185028i$$

Kvadrat ildizini topish bo'yicha qadamlar

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=\frac{2}{3}\times 3^{\frac{1}{2}}\left(3x^{2}+15\right)$$

\frac{2}{3}\times 3^{\frac{1}{2}} ga 3x^{2}+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}=2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}$$

Ikkala tarafdan 10\times 3^{\frac{1}{2}} ni ayirish.

$$2\times 3^{\frac{1}{2}}x^{2}=2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}-10\times 3^{\frac{1}{2}}$$

-\frac{28}{3}\times 3^{\frac{1}{2}} ni olish uchun \frac{2}{3}\times 3^{\frac{1}{2}} va -10\times 3^{\frac{1}{2}} ni birlashtirish.

$$2\times 3^{\frac{1}{2}}x^{2}=2\sqrt{2}-\frac{28}{3}\times 3^{\frac{1}{2}}$$

Shartlarni qayta saralash.

$$2\sqrt{3}x^{2}=-\frac{28}{3}\sqrt{3}+2\sqrt{2}$$

2\sqrt{3} ga bo'lish 2\sqrt{3} ga ko'paytirishni bekor qiladi.

$$x^{2}=\frac{-\frac{28\sqrt{3}}{3}+2\sqrt{2}}{2\sqrt{3}}$$

-\frac{28\sqrt{3}}{3}+2\sqrt{2} ni 2\sqrt{3} ga bo'lish.

$$x^{2}=\frac{\sqrt{6}-14}{3}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3} x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=\frac{2}{3}\times 3^{\frac{1}{2}}\left(3x^{2}+15\right)$$

\frac{2}{3}\times 3^{\frac{1}{2}} ga 3x^{2}+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}=2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}$$

Ikkala tarafdan 2\sqrt{2} ni ayirish.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}-2\sqrt{2}=\frac{2}{3}\times 3^{\frac{1}{2}}$$

Ikkala tarafdan \frac{2}{3}\times 3^{\frac{1}{2}} ni ayirish.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}-2\sqrt{2}-\frac{2}{3}\times 3^{\frac{1}{2}}=0$$

\frac{28}{3}\times 3^{\frac{1}{2}} ni olish uchun 10\times 3^{\frac{1}{2}} va -\frac{2}{3}\times 3^{\frac{1}{2}} ni birlashtirish.

$$2\times 3^{\frac{1}{2}}x^{2}+\frac{28}{3}\times 3^{\frac{1}{2}}-2\sqrt{2}=0$$

Shartlarni qayta saralash.

$$2\sqrt{3}x^{2}-2\sqrt{2}+\frac{28}{3}\sqrt{3}=0$$

Bu kabi kvadrat tenglamalarni x^{2} sharti bilan, biroq x shartisiz hamon kvadrat tenglamasidan foydalanib yechish mumkin, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, ular standart formulaga solingandan so'ng: ax^{2}+bx+c=0.

$$2\sqrt{3}x^{2}+\frac{28\sqrt{3}}{3}-2\sqrt{2}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2\sqrt{3} ni a, 0 ni b va -2\sqrt{2}+\frac{28\sqrt{3}}{3} ni c bilan almashtiring.

$$x=\frac{0±\sqrt{0^{2}-4\times 2\sqrt{3}\left(\frac{28\sqrt{3}}{3}-2\sqrt{2}\right)}}{2\times 2\sqrt{3}}$$

0 kvadratini chiqarish.

$$x=\frac{0±\sqrt{-4\times 2\sqrt{3}\left(\frac{28\sqrt{3}}{3}-2\sqrt{2}\right)}}{2\times 2\sqrt{3}}$$

-4 ni 2\sqrt{3} marotabaga ko'paytirish.

$$x=\frac{0±\sqrt{\left(-8\sqrt{3}\right)\left(\frac{28\sqrt{3}}{3}-2\sqrt{2}\right)}}{2\times 2\sqrt{3}}$$

-8\sqrt{3} ni -2\sqrt{2}+\frac{28\sqrt{3}}{3} marotabaga ko'paytirish.

$$x=\frac{0±\sqrt{16\sqrt{6}-224}}{2\times 2\sqrt{3}}$$

16\sqrt{6}-224 ning kvadrat ildizini chiqarish.

$$x=\frac{0±4i\sqrt{14-\sqrt{6}}}{2\times 2\sqrt{3}}$$

2 ni 2\sqrt{3} marotabaga ko'paytirish.

$$x=\frac{0±4i\sqrt{14-\sqrt{6}}}{4\sqrt{3}}$$

x=\frac{0±4i\sqrt{14-\sqrt{6}}}{4\sqrt{3}} tenglamasini yeching, bunda ± musbat.

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3}$$

x=\frac{0±4i\sqrt{14-\sqrt{6}}}{4\sqrt{3}} tenglamasini yeching, bunda ± manfiy.

$$x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}$$

Tenglama yechildi.

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3} x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}$$