3:7=(x+1):2x eching

$$3 : 7 = ( x + 1 ) : 2 x$$

x uchun yechish

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2}\approx 0,552208562$$

$$x=-\frac{\sqrt{217}}{14}-\frac{1}{2}\approx -1,552208562$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 14 ga, 7,2 ning eng kichik karralisiga ko‘paytiring.

$$6=7\left(x+1\right)x$$

7 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6=\left(7x+7\right)x$$

7x+7 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6=7x^{2}+7x$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$7x^{2}+7x=6$$

Ikkala tarafdan 6 ni ayirish.

$$7x^{2}+7x-6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, 7 ni b va -6 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 7\left(-6\right)}}{2\times 7}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 7\left(-6\right)}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-28\left(-6\right)}}{2\times 7}$$

-28 ni -6 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+168}}{2\times 7}$$

49 ni 168 ga qo'shish.

$$x=\frac{-7±\sqrt{217}}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{217}}{14}$$

x=\frac{-7±\sqrt{217}}{14} tenglamasini yeching, bunda ± musbat. -7 ni \sqrt{217} ga qo'shish.

$$x=\frac{\sqrt{217}-7}{14}$$

-7+\sqrt{217} ni 14 ga bo'lish.

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2}$$

x=\frac{-7±\sqrt{217}}{14} tenglamasini yeching, bunda ± manfiy. -7 dan \sqrt{217} ni ayirish.

$$x=\frac{-\sqrt{217}-7}{14}$$

-7-\sqrt{217} ni 14 ga bo'lish.

$$x=-\frac{\sqrt{217}}{14}-\frac{1}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2} x=-\frac{\sqrt{217}}{14}-\frac{1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 14 ga, 7,2 ning eng kichik karralisiga ko‘paytiring.

$$6=7\left(x+1\right)x$$

7 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6=\left(7x+7\right)x$$

7x+7 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6=7x^{2}+7x$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$7x^{2}+7x=6$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}+7x}{7}=\frac{6}{7}$$

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{7}{7}x=\frac{6}{7}$$

7 ni 7 ga bo'lish.

$$x^{2}+x=\frac{6}{7}$$

1 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{2} olish uchun. Keyin, \frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{6}{7}+\left(\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$x^{2}+x+\frac{1}{4}=\frac{6}{7}+\frac{1}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{6}{7} ni \frac{1}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+x+\frac{1}{4}=\frac{31}{28}$$

x^{2}+x+\frac{1}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{31}{28}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{31}{28}}$$

Qisqartirish.

$$x+\frac{1}{2}=\frac{\sqrt{217}}{14} x+\frac{1}{2}=-\frac{\sqrt{217}}{14}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2} x=-\frac{\sqrt{217}}{14}-\frac{1}{2}$$