4z^2+60z=800 eching

$$4 z ^ { 2 } + 60 z = 800$$

z uchun yechish

$$z = \frac{5 \sqrt{41} - 15}{2} \approx 8,507810594$$

$$z=\frac{-5\sqrt{41}-15}{2}\approx -23,507810594$$

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.

$$4z^{2}+60z=800$$

Tenglamaning ikkala tarafidan 800 ni ayirish.

$$4z^{2}+60z-800=800-800$$

O‘zidan 800 ayirilsa 0 qoladi.

$$4z^{2}+60z-800=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 60 ni b va -800 ni c bilan almashtiring.

$$z=\frac{-60±\sqrt{60^{2}-4\times 4\left(-800\right)}}{2\times 4}$$

60 kvadratini chiqarish.

$$z=\frac{-60±\sqrt{3600-4\times 4\left(-800\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$z=\frac{-60±\sqrt{3600-16\left(-800\right)}}{2\times 4}$$

-16 ni -800 marotabaga ko'paytirish.

$$z=\frac{-60±\sqrt{3600+12800}}{2\times 4}$$

3600 ni 12800 ga qo'shish.

$$z=\frac{-60±\sqrt{16400}}{2\times 4}$$

16400 ning kvadrat ildizini chiqarish.

$$z=\frac{-60±20\sqrt{41}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$z=\frac{-60±20\sqrt{41}}{8}$$

z=\frac{-60±20\sqrt{41}}{8} tenglamasini yeching, bunda ± musbat. -60 ni 20\sqrt{41} ga qo'shish.

$$z=\frac{20\sqrt{41}-60}{8}$$

-60+20\sqrt{41} ni 8 ga bo'lish.

$$z=\frac{5\sqrt{41}-15}{2}$$

z=\frac{-60±20\sqrt{41}}{8} tenglamasini yeching, bunda ± manfiy. -60 dan 20\sqrt{41} ni ayirish.

$$z=\frac{-20\sqrt{41}-60}{8}$$

-60-20\sqrt{41} ni 8 ga bo'lish.

$$z=\frac{-5\sqrt{41}-15}{2}$$

Tenglama yechildi.

$$z=\frac{5\sqrt{41}-15}{2} z=\frac{-5\sqrt{41}-15}{2}$$

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.

$$4z^{2}+60z=800$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4z^{2}+60z}{4}=\frac{800}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$z^{2}+\frac{60}{4}z=\frac{800}{4}$$

60 ni 4 ga bo'lish.

$$z^{2}+15z=\frac{800}{4}$$

800 ni 4 ga bo'lish.

$$z^{2}+15z=200$$

15 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{15}{2} olish uchun. Keyin, \frac{15}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$z^{2}+15z+\left(\frac{15}{2}\right)^{2}=200+\left(\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{15}{2} kvadratini chiqarish.

$$z^{2}+15z+\frac{225}{4}=200+\frac{225}{4}$$

200 ni \frac{225}{4} ga qo'shish.

$$z^{2}+15z+\frac{225}{4}=\frac{1025}{4}$$

z^{2}+15z+\frac{225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(z+\frac{15}{2}\right)^{2}=\frac{1025}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(z+\frac{15}{2}\right)^{2}}=\sqrt{\frac{1025}{4}}$$

Qisqartirish.

$$z+\frac{15}{2}=\frac{5\sqrt{41}}{2} z+\frac{15}{2}=-\frac{5\sqrt{41}}{2}$$

Tenglamaning ikkala tarafidan \frac{15}{2} ni ayirish.

$$z=\frac{5\sqrt{41}-15}{2} z=\frac{-5\sqrt{41}-15}{2}$$