-5z^2-4z+3=0 eching

$$- 5 z ^ { 2 } - 4 z + 3 = 0$$

z uchun yechish

$$z=\frac{\sqrt{19}-2}{5}\approx 0,471779789$$

$$z=\frac{-\sqrt{19}-2}{5}\approx -1,271779789$$

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.

$$-5z^{2}-4z+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, -4 ni b va 3 ni c bilan almashtiring.

$$z=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-5\right)\times 3}}{2\left(-5\right)}$$

-4 kvadratini chiqarish.

$$z=\frac{-\left(-4\right)±\sqrt{16-4\left(-5\right)\times 3}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$z=\frac{-\left(-4\right)±\sqrt{16+20\times 3}}{2\left(-5\right)}$$

20 ni 3 marotabaga ko'paytirish.

$$z=\frac{-\left(-4\right)±\sqrt{16+60}}{2\left(-5\right)}$$

16 ni 60 ga qo'shish.

$$z=\frac{-\left(-4\right)±\sqrt{76}}{2\left(-5\right)}$$

76 ning kvadrat ildizini chiqarish.

$$z=\frac{-\left(-4\right)±2\sqrt{19}}{2\left(-5\right)}$$

-4 ning teskarisi 4 ga teng.

$$z=\frac{4±2\sqrt{19}}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$z=\frac{4±2\sqrt{19}}{-10}$$

z=\frac{4±2\sqrt{19}}{-10} tenglamasini yeching, bunda ± musbat. 4 ni 2\sqrt{19} ga qo'shish.

$$z=\frac{2\sqrt{19}+4}{-10}$$

4+2\sqrt{19} ni -10 ga bo'lish.

$$z=\frac{-\sqrt{19}-2}{5}$$

z=\frac{4±2\sqrt{19}}{-10} tenglamasini yeching, bunda ± manfiy. 4 dan 2\sqrt{19} ni ayirish.

$$z=\frac{4-2\sqrt{19}}{-10}$$

4-2\sqrt{19} ni -10 ga bo'lish.

$$z=\frac{\sqrt{19}-2}{5}$$

Tenglama yechildi.

$$z=\frac{-\sqrt{19}-2}{5} z=\frac{\sqrt{19}-2}{5}$$

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.

$$-5z^{2}-4z+3=0$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$-5z^{2}-4z+3-3=-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$-5z^{2}-4z=-3$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5z^{2}-4z}{-5}=-\frac{3}{-5}$$

-5 ga bo'lish -5 ga ko'paytirishni bekor qiladi.

$$z^{2}+\left(-\frac{4}{-5}\right)z=-\frac{3}{-5}$$

-4 ni -5 ga bo'lish.

$$z^{2}+\frac{4}{5}z=-\frac{3}{-5}$$

-3 ni -5 ga bo'lish.

$$z^{2}+\frac{4}{5}z=\frac{3}{5}$$

\frac{4}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{2}{5} olish uchun. Keyin, \frac{2}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$z^{2}+\frac{4}{5}z+\left(\frac{2}{5}\right)^{2}=\frac{3}{5}+\left(\frac{2}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{2}{5} kvadratini chiqarish.

$$z^{2}+\frac{4}{5}z+\frac{4}{25}=\frac{3}{5}+\frac{4}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{5} ni \frac{4}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$z^{2}+\frac{4}{5}z+\frac{4}{25}=\frac{19}{25}$$

z^{2}+\frac{4}{5}z+\frac{4}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(z+\frac{2}{5}\right)^{2}=\frac{19}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(z+\frac{2}{5}\right)^{2}}=\sqrt{\frac{19}{25}}$$

Qisqartirish.

$$z+\frac{2}{5}=\frac{\sqrt{19}}{5} z+\frac{2}{5}=-\frac{\sqrt{19}}{5}$$

Tenglamaning ikkala tarafidan \frac{2}{5} ni ayirish.

$$z=\frac{\sqrt{19}-2}{5} z=\frac{-\sqrt{19}-2}{5}$$