1530quadx^2-30x-470=0 eching

$$1530 \quad x ^ { 2 } - 30 x - 470 = 0$$

x uchun yechish

$$x=\frac{\sqrt{3197}+1}{102}\approx 0,564137449$$

$$x=\frac{1-\sqrt{3197}}{102}\approx -0,544529606$$

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.

$$1530x^{2}-30x-470=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1530 ni a, -30 ni b va -470 ni c bilan almashtiring.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 1530\left(-470\right)}}{2\times 1530}$$

-30 kvadratini chiqarish.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 1530\left(-470\right)}}{2\times 1530}$$

-4 ni 1530 marotabaga ko'paytirish.

$$x=\frac{-\left(-30\right)±\sqrt{900-6120\left(-470\right)}}{2\times 1530}$$

-6120 ni -470 marotabaga ko'paytirish.

$$x=\frac{-\left(-30\right)±\sqrt{900+2876400}}{2\times 1530}$$

900 ni 2876400 ga qo'shish.

$$x=\frac{-\left(-30\right)±\sqrt{2877300}}{2\times 1530}$$

2877300 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-30\right)±30\sqrt{3197}}{2\times 1530}$$

-30 ning teskarisi 30 ga teng.

$$x=\frac{30±30\sqrt{3197}}{2\times 1530}$$

2 ni 1530 marotabaga ko'paytirish.

$$x=\frac{30±30\sqrt{3197}}{3060}$$

x=\frac{30±30\sqrt{3197}}{3060} tenglamasini yeching, bunda ± musbat. 30 ni 30\sqrt{3197} ga qo'shish.

$$x=\frac{30\sqrt{3197}+30}{3060}$$

30+30\sqrt{3197} ni 3060 ga bo'lish.

$$x=\frac{\sqrt{3197}+1}{102}$$

x=\frac{30±30\sqrt{3197}}{3060} tenglamasini yeching, bunda ± manfiy. 30 dan 30\sqrt{3197} ni ayirish.

$$x=\frac{30-30\sqrt{3197}}{3060}$$

30-30\sqrt{3197} ni 3060 ga bo'lish.

$$x=\frac{1-\sqrt{3197}}{102}$$

Tenglama yechildi.

$$x=\frac{\sqrt{3197}+1}{102} x=\frac{1-\sqrt{3197}}{102}$$

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.

$$1530x^{2}-30x-470=0$$

470 ni tenglamaning ikkala tarafiga qo'shish.

$$1530x^{2}-30x-470-\left(-470\right)=-\left(-470\right)$$

O‘zidan -470 ayirilsa 0 qoladi.

$$1530x^{2}-30x=-\left(-470\right)$$

0 dan -470 ni ayirish.

$$1530x^{2}-30x=470$$

Ikki tarafini 1530 ga bo‘ling.

$$\frac{1530x^{2}-30x}{1530}=\frac{470}{1530}$$

1530 ga bo'lish 1530 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{30}{1530}\right)x=\frac{470}{1530}$$

\frac{-30}{1530} ulushini 30 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{51}x=\frac{470}{1530}$$

\frac{470}{1530} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{51}x=\frac{47}{153}$$

-\frac{1}{51} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{102} olish uchun. Keyin, -\frac{1}{102} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{51}x+\left(-\frac{1}{102}\right)^{2}=\frac{47}{153}+\left(-\frac{1}{102}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{102} kvadratini chiqarish.

$$x^{2}-\frac{1}{51}x+\frac{1}{10404}=\frac{47}{153}+\frac{1}{10404}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{47}{153} ni \frac{1}{10404} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{51}x+\frac{1}{10404}=\frac{3197}{10404}$$

x^{2}-\frac{1}{51}x+\frac{1}{10404} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{102}\right)^{2}=\frac{3197}{10404}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{102}\right)^{2}}=\sqrt{\frac{3197}{10404}}$$

Qisqartirish.

$$x-\frac{1}{102}=\frac{\sqrt{3197}}{102} x-\frac{1}{102}=-\frac{\sqrt{3197}}{102}$$

\frac{1}{102} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{3197}+1}{102} x=\frac{1-\sqrt{3197}}{102}$$