112=6x-1/2*75x^2 eching

$$112 = 6 x - \frac { 1 } { 2 } \times 75 x ^ { 2 }$$

x uchun yechish (complex solution)

$$x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25}\approx 0,08+1,726344886i$$

$$x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25}\approx 0,08-1,726344886i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{75}{2} hosil qilish uchun \frac{1}{2} va 75 ni ko'paytirish.

$$112=6x-\frac{75}{2}x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$6x-\frac{75}{2}x^{2}=112$$

Ikkala tarafdan 112 ni ayirish.

$$6x-\frac{75}{2}x^{2}-112=0$$

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.

$$-\frac{75}{2}x^{2}+6x-112=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -\frac{75}{2} ni a, 6 ni b va -112 ni c bilan almashtiring.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-\frac{75}{2}\right)\left(-112\right)}}{2\left(-\frac{75}{2}\right)}$$

6 kvadratini chiqarish.

$$x=\frac{-6±\sqrt{36-4\left(-\frac{75}{2}\right)\left(-112\right)}}{2\left(-\frac{75}{2}\right)}$$

-4 ni -\frac{75}{2} marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+150\left(-112\right)}}{2\left(-\frac{75}{2}\right)}$$

150 ni -112 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36-16800}}{2\left(-\frac{75}{2}\right)}$$

36 ni -16800 ga qo'shish.

$$x=\frac{-6±\sqrt{-16764}}{2\left(-\frac{75}{2}\right)}$$

-16764 ning kvadrat ildizini chiqarish.

$$x=\frac{-6±2\sqrt{4191}i}{2\left(-\frac{75}{2}\right)}$$

2 ni -\frac{75}{2} marotabaga ko'paytirish.

$$x=\frac{-6±2\sqrt{4191}i}{-75}$$

x=\frac{-6±2\sqrt{4191}i}{-75} tenglamasini yeching, bunda ± musbat. -6 ni 2i\sqrt{4191} ga qo'shish.

$$x=\frac{-6+2\sqrt{4191}i}{-75}$$

-6+2i\sqrt{4191} ni -75 ga bo'lish.

$$x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$

x=\frac{-6±2\sqrt{4191}i}{-75} tenglamasini yeching, bunda ± manfiy. -6 dan 2i\sqrt{4191} ni ayirish.

$$x=\frac{-2\sqrt{4191}i-6}{-75}$$

-6-2i\sqrt{4191} ni -75 ga bo'lish.

$$x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$

Tenglama yechildi.

$$x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25} x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{75}{2} hosil qilish uchun \frac{1}{2} va 75 ni ko'paytirish.

$$112=6x-\frac{75}{2}x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$6x-\frac{75}{2}x^{2}=112$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-\frac{75}{2}x^{2}+6x=112$$

Tenglamaning ikki tarafini -\frac{75}{2} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$\frac{-\frac{75}{2}x^{2}+6x}{-\frac{75}{2}}=\frac{112}{-\frac{75}{2}}$$

-\frac{75}{2} ga bo'lish -\frac{75}{2} ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{6}{-\frac{75}{2}}x=\frac{112}{-\frac{75}{2}}$$

6 ni -\frac{75}{2} ga bo'lish 6 ga k'paytirish -\frac{75}{2} ga qaytarish.

$$x^{2}-\frac{4}{25}x=\frac{112}{-\frac{75}{2}}$$

112 ni -\frac{75}{2} ga bo'lish 112 ga k'paytirish -\frac{75}{2} ga qaytarish.

$$x^{2}-\frac{4}{25}x=-\frac{224}{75}$$

-\frac{4}{25} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{2}{25} olish uchun. Keyin, -\frac{2}{25} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{4}{25}x+\left(-\frac{2}{25}\right)^{2}=-\frac{224}{75}+\left(-\frac{2}{25}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{25} kvadratini chiqarish.

$$x^{2}-\frac{4}{25}x+\frac{4}{625}=-\frac{224}{75}+\frac{4}{625}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{224}{75} ni \frac{4}{625} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{4}{25}x+\frac{4}{625}=-\frac{5588}{1875}$$

x^{2}-\frac{4}{25}x+\frac{4}{625} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{2}{25}\right)^{2}=-\frac{5588}{1875}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2}{25}\right)^{2}}=\sqrt{-\frac{5588}{1875}}$$

Qisqartirish.

$$x-\frac{2}{25}=\frac{2\sqrt{4191}i}{75} x-\frac{2}{25}=-\frac{2\sqrt{4191}i}{75}$$

\frac{2}{25} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25} x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$