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4z^{2}+60z=600
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-600=600-600
Tenglamaning ikkala tarafidan 600 ni ayirish.
4z^{2}+60z-600=0
O‘zidan 600 ayirilsa 0 qoladi.
z=\frac{-60±\sqrt{60^{2}-4\times 4\left(-600\right)}}{2\times 4}
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 -600 ni c bilan almashtiring.
z=\frac{-60±\sqrt{3600-4\times 4\left(-600\right)}}{2\times 4}
60 kvadratini chiqarish.
z=\frac{-60±\sqrt{3600-16\left(-600\right)}}{2\times 4}
-4 ni 4 marotabaga ko'paytirish.
z=\frac{-60±\sqrt{3600+9600}}{2\times 4}
-16 ni -600 marotabaga ko'paytirish.
z=\frac{-60±\sqrt{13200}}{2\times 4}
3600 ni 9600 ga qo'shish.
z=\frac{-60±20\sqrt{33}}{2\times 4}
13200 ning kvadrat ildizini chiqarish.
z=\frac{-60±20\sqrt{33}}{8}
2 ni 4 marotabaga ko'paytirish.
z=\frac{20\sqrt{33}-60}{8}
z=\frac{-60±20\sqrt{33}}{8} tenglamasini yeching, bunda ± musbat. -60 ni 20\sqrt{33} ga qo'shish.
z=\frac{5\sqrt{33}-15}{2}
-60+20\sqrt{33} ni 8 ga bo'lish.
z=\frac{-20\sqrt{33}-60}{8}
z=\frac{-60±20\sqrt{33}}{8} tenglamasini yeching, bunda ± manfiy. -60 dan 20\sqrt{33} ni ayirish.
z=\frac{-5\sqrt{33}-15}{2}
-60-20\sqrt{33} ni 8 ga bo'lish.
z=\frac{5\sqrt{33}-15}{2} z=\frac{-5\sqrt{33}-15}{2}
Tenglama yechildi.
4z^{2}+60z=600
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
\frac{4z^{2}+60z}{4}=\frac{600}{4}
Ikki tarafini 4 ga bo‘ling.
z^{2}+\frac{60}{4}z=\frac{600}{4}
4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.
z^{2}+15z=\frac{600}{4}
60 ni 4 ga bo'lish.
z^{2}+15z=150
600 ni 4 ga bo'lish.
z^{2}+15z+\left(\frac{15}{2}\right)^{2}=150+\left(\frac{15}{2}\right)^{2}
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+\frac{225}{4}=150+\frac{225}{4}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{15}{2} kvadratini chiqarish.
z^{2}+15z+\frac{225}{4}=\frac{825}{4}
150 ni \frac{225}{4} ga qo'shish.
\left(z+\frac{15}{2}\right)^{2}=\frac{825}{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.
\sqrt{\left(z+\frac{15}{2}\right)^{2}}=\sqrt{\frac{825}{4}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
z+\frac{15}{2}=\frac{5\sqrt{33}}{2} z+\frac{15}{2}=-\frac{5\sqrt{33}}{2}
Qisqartirish.
z=\frac{5\sqrt{33}-15}{2} z=\frac{-5\sqrt{33}-15}{2}
Tenglamaning ikkala tarafidan \frac{15}{2} ni ayirish.