49x^2-70x+25=0 eching

$$49 x ^ { 2 } - 70 x + 25 = 0$$

x uchun yechish

$$x=\frac{5}{7}\approx 0,714285714$$

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.

$$49x^{2}-70x+25=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 49 ni a, -70 ni b va 25 ni c bilan almashtiring.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 49\times 25}}{2\times 49}$$

-70 kvadratini chiqarish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 49\times 25}}{2\times 49}$$

-4 ni 49 marotabaga ko'paytirish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-196\times 25}}{2\times 49}$$

-196 ni 25 marotabaga ko'paytirish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4900}}{2\times 49}$$

4900 ni -4900 ga qo'shish.

$$x=\frac{-\left(-70\right)±\sqrt{0}}{2\times 49}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{-70}{2\times 49}$$

-70 ning teskarisi 70 ga teng.

$$x=\frac{70}{2\times 49}$$

2 ni 49 marotabaga ko'paytirish.

$$x=\frac{70}{98}$$

\frac{70}{98} ulushini 14 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{5}{7}$$

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.

$$49x^{2}-70x+25=0$$

Tenglamaning ikkala tarafidan 25 ni ayirish.

$$49x^{2}-70x+25-25=-25$$

O‘zidan 25 ayirilsa 0 qoladi.

$$49x^{2}-70x=-25$$

Ikki tarafini 49 ga bo‘ling.

$$\frac{49x^{2}-70x}{49}=-\frac{25}{49}$$

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

$$x^{2}+\left(-\frac{70}{49}\right)x=-\frac{25}{49}$$

\frac{-70}{49} ulushini 7 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{10}{7}x=-\frac{25}{49}$$

-\frac{10}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{7} olish uchun. Keyin, -\frac{5}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{10}{7}x+\left(-\frac{5}{7}\right)^{2}=-\frac{25}{49}+\left(-\frac{5}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{7} kvadratini chiqarish.

$$x^{2}-\frac{10}{7}x+\frac{25}{49}=\frac{-25+25}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{25}{49} ni \frac{25}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{10}{7}x+\frac{25}{49}=0$$

x^{2}-\frac{10}{7}x+\frac{25}{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.

$$\left(x-\frac{5}{7}\right)^{2}=0$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{7}\right)^{2}}=\sqrt{0}$$

Qisqartirish.

$$x-\frac{5}{7}=0 x-\frac{5}{7}=0$$

\frac{5}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5}{7} x=\frac{5}{7}$$

Tenglama yechildi. Yechimlar bir xil.

$$x=\frac{5}{7}$$