97000=2288x^2+5873x+5440 eching

$$97000=2288 { x }^{ 2 } +5873x+5440$$

x uchun yechish

$$x = \frac{\sqrt{872449249} - 5873}{4576} \approx 5,171383716$$

$$x=\frac{-\sqrt{872449249}-5873}{4576}\approx -7,738254346$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$2288x^{2}+5873x+5440=97000$$

Ikkala tarafdan 97000 ni ayirish.

$$2288x^{2}+5873x+5440-97000=0$$

-91560 olish uchun 5440 dan 97000 ni ayirish.

$$2288x^{2}+5873x-91560=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2288 ni a, 5873 ni b va -91560 ni c bilan almashtiring.

$$x=\frac{-5873±\sqrt{5873^{2}-4\times 2288\left(-91560\right)}}{2\times 2288}$$

5873 kvadratini chiqarish.

$$x=\frac{-5873±\sqrt{34492129-4\times 2288\left(-91560\right)}}{2\times 2288}$$

-4 ni 2288 marotabaga ko'paytirish.

$$x=\frac{-5873±\sqrt{34492129-9152\left(-91560\right)}}{2\times 2288}$$

-9152 ni -91560 marotabaga ko'paytirish.

$$x=\frac{-5873±\sqrt{34492129+837957120}}{2\times 2288}$$

34492129 ni 837957120 ga qo'shish.

$$x=\frac{-5873±\sqrt{872449249}}{2\times 2288}$$

2 ni 2288 marotabaga ko'paytirish.

$$x=\frac{-5873±\sqrt{872449249}}{4576}$$

x=\frac{-5873±\sqrt{872449249}}{4576} tenglamasini yeching, bunda ± musbat. -5873 ni \sqrt{872449249} ga qo'shish.

$$x=\frac{\sqrt{872449249}-5873}{4576}$$

x=\frac{-5873±\sqrt{872449249}}{4576} tenglamasini yeching, bunda ± manfiy. -5873 dan \sqrt{872449249} ni ayirish.

$$x=\frac{-\sqrt{872449249}-5873}{4576}$$

Tenglama yechildi.

$$x=\frac{\sqrt{872449249}-5873}{4576} x=\frac{-\sqrt{872449249}-5873}{4576}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$2288x^{2}+5873x+5440=97000$$

Ikkala tarafdan 5440 ni ayirish.

$$2288x^{2}+5873x=97000-5440$$

91560 olish uchun 97000 dan 5440 ni ayirish.

$$2288x^{2}+5873x=91560$$

Ikki tarafini 2288 ga bo‘ling.

$$\frac{2288x^{2}+5873x}{2288}=\frac{91560}{2288}$$

2288 ga bo'lish 2288 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{5873}{2288}x=\frac{91560}{2288}$$

\frac{91560}{2288} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{5873}{2288}x=\frac{11445}{286}$$

\frac{5873}{2288} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5873}{4576} olish uchun. Keyin, \frac{5873}{4576} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{5873}{2288}x+\left(\frac{5873}{4576}\right)^{2}=\frac{11445}{286}+\left(\frac{5873}{4576}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5873}{4576} kvadratini chiqarish.

$$x^{2}+\frac{5873}{2288}x+\frac{34492129}{20939776}=\frac{11445}{286}+\frac{34492129}{20939776}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{11445}{286} ni \frac{34492129}{20939776} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{5873}{2288}x+\frac{34492129}{20939776}=\frac{872449249}{20939776}$$

x^{2}+\frac{5873}{2288}x+\frac{34492129}{20939776} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{5873}{4576}\right)^{2}=\frac{872449249}{20939776}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5873}{4576}\right)^{2}}=\sqrt{\frac{872449249}{20939776}}$$

Qisqartirish.

$$x+\frac{5873}{4576}=\frac{\sqrt{872449249}}{4576} x+\frac{5873}{4576}=-\frac{\sqrt{872449249}}{4576}$$

Tenglamaning ikkala tarafidan \frac{5873}{4576} ni ayirish.

$$x=\frac{\sqrt{872449249}-5873}{4576} x=\frac{-\sqrt{872449249}-5873}{4576}$$