5=-1/60x^2+139/60x eching

$$5 = - \frac { 1 } { 60 } x ^ { 2 } + \frac { 139 } { 60 } x$$

x uchun yechish

$$x = \frac{\sqrt{18121} + 139}{2} \approx 136,807131866$$

$$x = \frac{139 - \sqrt{18121}}{2} \approx 2,192868134$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-\frac{1}{60}x^{2}+\frac{139}{60}x=5$$

Ikkala tarafdan 5 ni ayirish.

$$-\frac{1}{60}x^{2}+\frac{139}{60}x-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -\frac{1}{60} ni a, \frac{139}{60} ni b va -5 ni c bilan almashtiring.

$$x=\frac{-\frac{139}{60}±\sqrt{\left(\frac{139}{60}\right)^{2}-4\left(-\frac{1}{60}\right)\left(-5\right)}}{2\left(-\frac{1}{60}\right)}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{139}{60} kvadratini chiqarish.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{19321}{3600}-4\left(-\frac{1}{60}\right)\left(-5\right)}}{2\left(-\frac{1}{60}\right)}$$

-4 ni -\frac{1}{60} marotabaga ko'paytirish.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{19321}{3600}+\frac{1}{15}\left(-5\right)}}{2\left(-\frac{1}{60}\right)}$$

\frac{1}{15} ni -5 marotabaga ko'paytirish.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{19321}{3600}-\frac{1}{3}}}{2\left(-\frac{1}{60}\right)}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{19321}{3600} ni -\frac{1}{3} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\frac{139}{60}±\sqrt{\frac{18121}{3600}}}{2\left(-\frac{1}{60}\right)}$$

\frac{18121}{3600} ning kvadrat ildizini chiqarish.

$$x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{2\left(-\frac{1}{60}\right)}$$

2 ni -\frac{1}{60} marotabaga ko'paytirish.

$$x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{-\frac{1}{30}}$$

x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{-\frac{1}{30}} tenglamasini yeching, bunda ± musbat. -\frac{139}{60} ni \frac{\sqrt{18121}}{60} ga qo'shish.

$$x=\frac{\sqrt{18121}-139}{-\frac{1}{30}\times 60}$$

\frac{-139+\sqrt{18121}}{60} ni -\frac{1}{30} ga bo'lish \frac{-139+\sqrt{18121}}{60} ga k'paytirish -\frac{1}{30} ga qaytarish.

$$x=\frac{139-\sqrt{18121}}{2}$$

x=\frac{-\frac{139}{60}±\frac{\sqrt{18121}}{60}}{-\frac{1}{30}} tenglamasini yeching, bunda ± manfiy. -\frac{139}{60} dan \frac{\sqrt{18121}}{60} ni ayirish.

$$x=\frac{-\sqrt{18121}-139}{-\frac{1}{30}\times 60}$$

\frac{-139-\sqrt{18121}}{60} ni -\frac{1}{30} ga bo'lish \frac{-139-\sqrt{18121}}{60} ga k'paytirish -\frac{1}{30} ga qaytarish.

$$x=\frac{\sqrt{18121}+139}{2}$$

Tenglama yechildi.

$$x=\frac{139-\sqrt{18121}}{2} x=\frac{\sqrt{18121}+139}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-\frac{1}{60}x^{2}+\frac{139}{60}x=5$$

Ikkala tarafini -60 ga ko‘paytiring.

$$\frac{-\frac{1}{60}x^{2}+\frac{139}{60}x}{-\frac{1}{60}}=\frac{5}{-\frac{1}{60}}$$

-\frac{1}{60} ga bo'lish -\frac{1}{60} ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{\frac{139}{60}}{-\frac{1}{60}}x=\frac{5}{-\frac{1}{60}}$$

\frac{139}{60} ni -\frac{1}{60} ga bo'lish \frac{139}{60} ga k'paytirish -\frac{1}{60} ga qaytarish.

$$x^{2}-139x=\frac{5}{-\frac{1}{60}}$$

5 ni -\frac{1}{60} ga bo'lish 5 ga k'paytirish -\frac{1}{60} ga qaytarish.

$$x^{2}-139x=-300$$

-139 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{139}{2} olish uchun. Keyin, -\frac{139}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-139x+\left(-\frac{139}{2}\right)^{2}=-300+\left(-\frac{139}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{139}{2} kvadratini chiqarish.

$$x^{2}-139x+\frac{19321}{4}=-300+\frac{19321}{4}$$

-300 ni \frac{19321}{4} ga qo'shish.

$$x^{2}-139x+\frac{19321}{4}=\frac{18121}{4}$$

x^{2}-139x+\frac{19321}{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.

$$\left(x-\frac{139}{2}\right)^{2}=\frac{18121}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{139}{2}\right)^{2}}=\sqrt{\frac{18121}{4}}$$

Qisqartirish.

$$x-\frac{139}{2}=\frac{\sqrt{18121}}{2} x-\frac{139}{2}=-\frac{\sqrt{18121}}{2}$$

\frac{139}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{18121}+139}{2} x=\frac{139-\sqrt{18121}}{2}$$