z^2-25*10^-12z+16*10^-12=0 eching

$${ z }^{ 2 } -25 \times { 10 }^{ -12 } z+16 \times { 10 }^{ -12 } = 0$$

z uchun yechish

$$z=\frac{1+\sqrt{102399999999}i}{80000000000}\approx 1,25 \cdot 10^{-11}+0,000004i$$

$$z=\frac{-\sqrt{102399999999}i+1}{80000000000}\approx 1,25 \cdot 10^{-11}-0,000004i$$

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.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{62500000000}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -\frac{1}{40000000000} ni b va \frac{1}{62500000000} ni c bilan almashtiring.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\sqrt{\left(-\frac{1}{40000000000}\right)^{2}-4\times \frac{1}{62500000000}}}{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{40000000000} kvadratini chiqarish.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\sqrt{\frac{1}{1600000000000000000000}-4\times \frac{1}{62500000000}}}{2}$$

-4 ni \frac{1}{62500000000} marotabaga ko'paytirish.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\sqrt{\frac{1}{1600000000000000000000}-\frac{1}{15625000000}}}{2}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{1600000000000000000000} ni -\frac{1}{15625000000} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\sqrt{-\frac{102399999999}{1600000000000000000000}}}{2}$$

-\frac{102399999999}{1600000000000000000000} ning kvadrat ildizini chiqarish.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\frac{\sqrt{102399999999}i}{40000000000}}{2}$$

-\frac{1}{40000000000} ning teskarisi \frac{1}{40000000000} ga teng.

$$z=\frac{\frac{1}{40000000000}±\frac{\sqrt{102399999999}i}{40000000000}}{2}$$

z=\frac{\frac{1}{40000000000}±\frac{\sqrt{102399999999}i}{40000000000}}{2} tenglamasini yeching, bunda ± musbat. \frac{1}{40000000000} ni \frac{i\sqrt{102399999999}}{40000000000} ga qo'shish.

$$z=\frac{1+\sqrt{102399999999}i}{2\times 40000000000}$$

\frac{1+i\sqrt{102399999999}}{40000000000} ni 2 ga bo'lish.

$$z=\frac{1+\sqrt{102399999999}i}{80000000000}$$

z=\frac{\frac{1}{40000000000}±\frac{\sqrt{102399999999}i}{40000000000}}{2} tenglamasini yeching, bunda ± manfiy. \frac{1}{40000000000} dan \frac{i\sqrt{102399999999}}{40000000000} ni ayirish.

$$z=\frac{-\sqrt{102399999999}i+1}{2\times 40000000000}$$

\frac{1-i\sqrt{102399999999}}{40000000000} ni 2 ga bo'lish.

$$z=\frac{-\sqrt{102399999999}i+1}{80000000000}$$

Tenglama yechildi.

$$z=\frac{1+\sqrt{102399999999}i}{80000000000} z=\frac{-\sqrt{102399999999}i+1}{80000000000}$$

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.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{62500000000}=0$$

Tenglamaning ikkala tarafidan \frac{1}{62500000000} ni ayirish.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{62500000000}-\frac{1}{62500000000}=-\frac{1}{62500000000}$$

O‘zidan \frac{1}{62500000000} ayirilsa 0 qoladi.

$$z^{2}-\frac{1}{40000000000}z=-\frac{1}{62500000000}$$

-\frac{1}{40000000000} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{80000000000} olish uchun. Keyin, -\frac{1}{80000000000} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$z^{2}-\frac{1}{40000000000}z+\left(-\frac{1}{80000000000}\right)^{2}=-\frac{1}{62500000000}+\left(-\frac{1}{80000000000}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{80000000000} kvadratini chiqarish.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{6400000000000000000000}=-\frac{1}{62500000000}+\frac{1}{6400000000000000000000}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{62500000000} ni \frac{1}{6400000000000000000000} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{6400000000000000000000}=-\frac{102399999999}{6400000000000000000000}$$

z^{2}-\frac{1}{40000000000}z+\frac{1}{6400000000000000000000} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(z-\frac{1}{80000000000}\right)^{2}=-\frac{102399999999}{6400000000000000000000}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(z-\frac{1}{80000000000}\right)^{2}}=\sqrt{-\frac{102399999999}{6400000000000000000000}}$$

Qisqartirish.

$$z-\frac{1}{80000000000}=\frac{\sqrt{102399999999}i}{80000000000} z-\frac{1}{80000000000}=-\frac{\sqrt{102399999999}i}{80000000000}$$

\frac{1}{80000000000} ni tenglamaning ikkala tarafiga qo'shish.

$$z=\frac{1+\sqrt{102399999999}i}{80000000000} z=\frac{-\sqrt{102399999999}i+1}{80000000000}$$