49x^2-84x+36=0 eching

$$49 x ^ { 2 } - 84 x + 36 = 0$$

x uchun yechish

$$x=\frac{6}{7}\approx 0,857142857$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$49x^{2}-84x+36=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 49 ni a, -84 ni b va 36 ni c bilan almashtiring.

$$x=\frac{-\left(-84\right)±\sqrt{\left(-84\right)^{2}-4\times 49\times 36}}{2\times 49}$$

-84 kvadratini chiqarish.

$$x=\frac{-\left(-84\right)±\sqrt{7056-4\times 49\times 36}}{2\times 49}$$

-4 ni 49 marotabaga ko'paytirish.

$$x=\frac{-\left(-84\right)±\sqrt{7056-196\times 36}}{2\times 49}$$

-196 ni 36 marotabaga ko'paytirish.

$$x=\frac{-\left(-84\right)±\sqrt{7056-7056}}{2\times 49}$$

7056 ni -7056 ga qo'shish.

$$x=\frac{-\left(-84\right)±\sqrt{0}}{2\times 49}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{-84}{2\times 49}$$

-84 ning teskarisi 84 ga teng.

$$x=\frac{84}{2\times 49}$$

2 ni 49 marotabaga ko'paytirish.

$$x=\frac{84}{98}$$

\frac{84}{98} ulushini 14 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{6}{7}$$

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$49x^{2}-84x+36=0$$

Tenglamaning ikkala tarafidan 36 ni ayirish.

$$49x^{2}-84x+36-36=-36$$

O‘zidan 36 ayirilsa 0 qoladi.

$$49x^{2}-84x=-36$$

Ikki tarafini 49 ga bo‘ling.

$$\frac{49x^{2}-84x}{49}=-\frac{36}{49}$$

49 ga bo'lish 49 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{84}{49}\right)x=-\frac{36}{49}$$

\frac{-84}{49} ulushini 7 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{12}{7}x=-\frac{36}{49}$$

-\frac{12}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{6}{7} olish uchun. Keyin, -\frac{6}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{12}{7}x+\left(-\frac{6}{7}\right)^{2}=-\frac{36}{49}+\left(-\frac{6}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{6}{7} kvadratini chiqarish.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=\frac{-36+36}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{36}{49} ni \frac{36}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=0$$

x^{2}-\frac{12}{7}x+\frac{36}{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.

$$\left(x-\frac{6}{7}\right)^{2}=0$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{6}{7}\right)^{2}}=\sqrt{0}$$

Qisqartirish.

$$x-\frac{6}{7}=0 x-\frac{6}{7}=0$$

\frac{6}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{6}{7} x=\frac{6}{7}$$

Tenglama yechildi. Yechimlar bir xil.

$$x=\frac{6}{7}$$