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4x^{2}+4x=5
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.
4x^{2}+4x-5=5-5
Tenglamaning ikkala tarafidan 5 ni ayirish.
4x^{2}+4x-5=0
O‘zidan 5 ayirilsa 0 qoladi.
x=\frac{-4±\sqrt{4^{2}-4\times 4\left(-5\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, 4 ni b va -5 ni c bilan almashtiring.
x=\frac{-4±\sqrt{16-4\times 4\left(-5\right)}}{2\times 4}
4 kvadratini chiqarish.
x=\frac{-4±\sqrt{16-16\left(-5\right)}}{2\times 4}
-4 ni 4 marotabaga ko'paytirish.
x=\frac{-4±\sqrt{16+80}}{2\times 4}
-16 ni -5 marotabaga ko'paytirish.
x=\frac{-4±\sqrt{96}}{2\times 4}
16 ni 80 ga qo'shish.
x=\frac{-4±4\sqrt{6}}{2\times 4}
96 ning kvadrat ildizini chiqarish.
x=\frac{-4±4\sqrt{6}}{8}
2 ni 4 marotabaga ko'paytirish.
x=\frac{4\sqrt{6}-4}{8}
x=\frac{-4±4\sqrt{6}}{8} tenglamasini yeching, bunda ± musbat. -4 ni 4\sqrt{6} ga qo'shish.
x=\frac{\sqrt{6}-1}{2}
-4+4\sqrt{6} ni 8 ga bo'lish.
x=\frac{-4\sqrt{6}-4}{8}
x=\frac{-4±4\sqrt{6}}{8} tenglamasini yeching, bunda ± manfiy. -4 dan 4\sqrt{6} ni ayirish.
x=\frac{-\sqrt{6}-1}{2}
-4-4\sqrt{6} ni 8 ga bo'lish.
x=\frac{\sqrt{6}-1}{2} x=\frac{-\sqrt{6}-1}{2}
Tenglama yechildi.
4x^{2}+4x=5
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
\frac{4x^{2}+4x}{4}=\frac{5}{4}
Ikki tarafini 4 ga bo‘ling.
x^{2}+\frac{4}{4}x=\frac{5}{4}
4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.
x^{2}+x=\frac{5}{4}
4 ni 4 ga bo'lish.
x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{5}{4}+\left(\frac{1}{2}\right)^{2}
1 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{2} olish uchun. Keyin, \frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
x^{2}+x+\frac{1}{4}=\frac{5+1}{4}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.
x^{2}+x+\frac{1}{4}=\frac{3}{2}
Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{4} ni \frac{1}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.
\left(x+\frac{1}{2}\right)^{2}=\frac{3}{2}
x^{2}+x+\frac{1}{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(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{3}{2}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
x+\frac{1}{2}=\frac{\sqrt{6}}{2} x+\frac{1}{2}=-\frac{\sqrt{6}}{2}
Qisqartirish.
x=\frac{\sqrt{6}-1}{2} x=\frac{-\sqrt{6}-1}{2}
Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.