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3x^{2}+11x=-24
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.
3x^{2}+11x-\left(-24\right)=-24-\left(-24\right)
24 ni tenglamaning ikkala tarafiga qo'shish.
3x^{2}+11x-\left(-24\right)=0
O‘zidan -24 ayirilsa 0 qoladi.
3x^{2}+11x+24=0
0 dan -24 ni ayirish.
x=\frac{-11±\sqrt{11^{2}-4\times 3\times 24}}{2\times 3}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 11 ni b va 24 ni c bilan almashtiring.
x=\frac{-11±\sqrt{121-4\times 3\times 24}}{2\times 3}
11 kvadratini chiqarish.
x=\frac{-11±\sqrt{121-12\times 24}}{2\times 3}
-4 ni 3 marotabaga ko'paytirish.
x=\frac{-11±\sqrt{121-288}}{2\times 3}
-12 ni 24 marotabaga ko'paytirish.
x=\frac{-11±\sqrt{-167}}{2\times 3}
121 ni -288 ga qo'shish.
x=\frac{-11±\sqrt{167}i}{2\times 3}
-167 ning kvadrat ildizini chiqarish.
x=\frac{-11±\sqrt{167}i}{6}
2 ni 3 marotabaga ko'paytirish.
x=\frac{-11+\sqrt{167}i}{6}
x=\frac{-11±\sqrt{167}i}{6} tenglamasini yeching, bunda ± musbat. -11 ni i\sqrt{167} ga qo'shish.
x=\frac{-\sqrt{167}i-11}{6}
x=\frac{-11±\sqrt{167}i}{6} tenglamasini yeching, bunda ± manfiy. -11 dan i\sqrt{167} ni ayirish.
x=\frac{-11+\sqrt{167}i}{6} x=\frac{-\sqrt{167}i-11}{6}
Tenglama yechildi.
3x^{2}+11x=-24
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
\frac{3x^{2}+11x}{3}=-\frac{24}{3}
Ikki tarafini 3 ga bo‘ling.
x^{2}+\frac{11}{3}x=-\frac{24}{3}
3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.
x^{2}+\frac{11}{3}x=-8
-24 ni 3 ga bo'lish.
x^{2}+\frac{11}{3}x+\left(\frac{11}{6}\right)^{2}=-8+\left(\frac{11}{6}\right)^{2}
\frac{11}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{11}{6} olish uchun. Keyin, \frac{11}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
x^{2}+\frac{11}{3}x+\frac{121}{36}=-8+\frac{121}{36}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{11}{6} kvadratini chiqarish.
x^{2}+\frac{11}{3}x+\frac{121}{36}=-\frac{167}{36}
-8 ni \frac{121}{36} ga qo'shish.
\left(x+\frac{11}{6}\right)^{2}=-\frac{167}{36}
x^{2}+\frac{11}{3}x+\frac{121}{36} 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{11}{6}\right)^{2}}=\sqrt{-\frac{167}{36}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
x+\frac{11}{6}=\frac{\sqrt{167}i}{6} x+\frac{11}{6}=-\frac{\sqrt{167}i}{6}
Qisqartirish.
x=\frac{-11+\sqrt{167}i}{6} x=\frac{-\sqrt{167}i-11}{6}
Tenglamaning ikkala tarafidan \frac{11}{6} ni ayirish.