7x^2+2=30x-10 eching

$$7 { x }^{ 2 } +2=30x-10$$

x uchun yechish

$$x = \frac{\sqrt{141} + 15}{7} \approx 3,839191727$$

$$x=\frac{15-\sqrt{141}}{7}\approx 0,446522559$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 30x ni ayirish.

$$7x^{2}+2-30x=-10$$

10 ni ikki tarafga qo’shing.

$$7x^{2}+2-30x+10=0$$

12 olish uchun 2 va 10'ni qo'shing.

$$7x^{2}+12-30x=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$7x^{2}-30x+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, -30 ni b va 12 ni c bilan almashtiring.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 7\times 12}}{2\times 7}$$

-30 kvadratini chiqarish.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 7\times 12}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-30\right)±\sqrt{900-28\times 12}}{2\times 7}$$

-28 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-30\right)±\sqrt{900-336}}{2\times 7}$$

900 ni -336 ga qo'shish.

$$x=\frac{-\left(-30\right)±\sqrt{564}}{2\times 7}$$

564 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-30\right)±2\sqrt{141}}{2\times 7}$$

-30 ning teskarisi 30 ga teng.

$$x=\frac{30±2\sqrt{141}}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{30±2\sqrt{141}}{14}$$

x=\frac{30±2\sqrt{141}}{14} tenglamasini yeching, bunda ± musbat. 30 ni 2\sqrt{141} ga qo'shish.

$$x=\frac{2\sqrt{141}+30}{14}$$

30+2\sqrt{141} ni 14 ga bo'lish.

$$x=\frac{\sqrt{141}+15}{7}$$

x=\frac{30±2\sqrt{141}}{14} tenglamasini yeching, bunda ± manfiy. 30 dan 2\sqrt{141} ni ayirish.

$$x=\frac{30-2\sqrt{141}}{14}$$

30-2\sqrt{141} ni 14 ga bo'lish.

$$x=\frac{15-\sqrt{141}}{7}$$

Tenglama yechildi.

$$x=\frac{\sqrt{141}+15}{7} x=\frac{15-\sqrt{141}}{7}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 30x ni ayirish.

$$7x^{2}+2-30x=-10$$

Ikkala tarafdan 2 ni ayirish.

$$7x^{2}-30x=-10-2$$

-12 olish uchun -10 dan 2 ni ayirish.

$$7x^{2}-30x=-12$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}-30x}{7}=-\frac{12}{7}$$

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

$$x^{2}-\frac{30}{7}x=-\frac{12}{7}$$

-\frac{30}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{7} olish uchun. Keyin, -\frac{15}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{30}{7}x+\left(-\frac{15}{7}\right)^{2}=-\frac{12}{7}+\left(-\frac{15}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{7} kvadratini chiqarish.

$$x^{2}-\frac{30}{7}x+\frac{225}{49}=-\frac{12}{7}+\frac{225}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{12}{7} ni \frac{225}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{30}{7}x+\frac{225}{49}=\frac{141}{49}$$

x^{2}-\frac{30}{7}x+\frac{225}{49} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{15}{7}\right)^{2}=\frac{141}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{7}\right)^{2}}=\sqrt{\frac{141}{49}}$$

Qisqartirish.

$$x-\frac{15}{7}=\frac{\sqrt{141}}{7} x-\frac{15}{7}=-\frac{\sqrt{141}}{7}$$

\frac{15}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{141}+15}{7} x=\frac{15-\sqrt{141}}{7}$$