125x^2+x-12=19x eching

$$125 x ^ { 2 } + x - 12 = 19 x$$

x uchun yechish

$$x=\frac{\sqrt{1581}+9}{125}\approx 0,390094326$$

$$x=\frac{9-\sqrt{1581}}{125}\approx -0,246094326$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 19x ni ayirish.

$$125x^{2}+x-12-19x=0$$

-18x ni olish uchun x va -19x ni birlashtirish.

$$125x^{2}-18x-12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 125 ni a, -18 ni b va -12 ni c bilan almashtiring.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 125\left(-12\right)}}{2\times 125}$$

-18 kvadratini chiqarish.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 125\left(-12\right)}}{2\times 125}$$

-4 ni 125 marotabaga ko'paytirish.

$$x=\frac{-\left(-18\right)±\sqrt{324-500\left(-12\right)}}{2\times 125}$$

-500 ni -12 marotabaga ko'paytirish.

$$x=\frac{-\left(-18\right)±\sqrt{324+6000}}{2\times 125}$$

324 ni 6000 ga qo'shish.

$$x=\frac{-\left(-18\right)±\sqrt{6324}}{2\times 125}$$

6324 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-18\right)±2\sqrt{1581}}{2\times 125}$$

-18 ning teskarisi 18 ga teng.

$$x=\frac{18±2\sqrt{1581}}{2\times 125}$$

2 ni 125 marotabaga ko'paytirish.

$$x=\frac{18±2\sqrt{1581}}{250}$$

x=\frac{18±2\sqrt{1581}}{250} tenglamasini yeching, bunda ± musbat. 18 ni 2\sqrt{1581} ga qo'shish.

$$x=\frac{2\sqrt{1581}+18}{250}$$

18+2\sqrt{1581} ni 250 ga bo'lish.

$$x=\frac{\sqrt{1581}+9}{125}$$

x=\frac{18±2\sqrt{1581}}{250} tenglamasini yeching, bunda ± manfiy. 18 dan 2\sqrt{1581} ni ayirish.

$$x=\frac{18-2\sqrt{1581}}{250}$$

18-2\sqrt{1581} ni 250 ga bo'lish.

$$x=\frac{9-\sqrt{1581}}{125}$$

Tenglama yechildi.

$$x=\frac{\sqrt{1581}+9}{125} x=\frac{9-\sqrt{1581}}{125}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 19x ni ayirish.

$$125x^{2}+x-12-19x=0$$

-18x ni olish uchun x va -19x ni birlashtirish.

$$125x^{2}-18x-12=0$$

12 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$125x^{2}-18x=12$$

Ikki tarafini 125 ga bo‘ling.

$$\frac{125x^{2}-18x}{125}=\frac{12}{125}$$

125 ga bo'lish 125 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{18}{125}x=\frac{12}{125}$$

-\frac{18}{125} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{125} olish uchun. Keyin, -\frac{9}{125} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{18}{125}x+\left(-\frac{9}{125}\right)^{2}=\frac{12}{125}+\left(-\frac{9}{125}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{125} kvadratini chiqarish.

$$x^{2}-\frac{18}{125}x+\frac{81}{15625}=\frac{12}{125}+\frac{81}{15625}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{12}{125} ni \frac{81}{15625} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{18}{125}x+\frac{81}{15625}=\frac{1581}{15625}$$

x^{2}-\frac{18}{125}x+\frac{81}{15625} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{9}{125}\right)^{2}=\frac{1581}{15625}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{125}\right)^{2}}=\sqrt{\frac{1581}{15625}}$$

Qisqartirish.

$$x-\frac{9}{125}=\frac{\sqrt{1581}}{125} x-\frac{9}{125}=-\frac{\sqrt{1581}}{125}$$

\frac{9}{125} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{1581}+9}{125} x=\frac{9-\sqrt{1581}}{125}$$