(2x+16):(95-5x)=3x eching

$$( 2 x + 16 ) : ( 95 - 5 x ) = 3 x$$

x uchun yechish

$$x = \frac{\sqrt{79129} + 283}{30} \approx 18,809959118$$

$$x=\frac{283-\sqrt{79129}}{30}\approx 0,056707548$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 19 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 5\left(-x+19\right) ga ko'paytirish.

$$2x+16=3x\times 5\left(-x+19\right)$$

15 hosil qilish uchun 3 va 5 ni ko'paytirish.

$$2x+16=15x\left(-x+19\right)$$

15x ga -x+19 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x+16=-15x^{2}+285x$$

15x^{2} ni ikki tarafga qo’shing.

$$2x+16+15x^{2}=285x$$

Ikkala tarafdan 285x ni ayirish.

$$2x+16+15x^{2}-285x=0$$

-283x ni olish uchun 2x va -285x ni birlashtirish.

$$-283x+16+15x^{2}=0$$

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.

$$15x^{2}-283x+16=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 15 ni a, -283 ni b va 16 ni c bilan almashtiring.

$$x=\frac{-\left(-283\right)±\sqrt{\left(-283\right)^{2}-4\times 15\times 16}}{2\times 15}$$

-283 kvadratini chiqarish.

$$x=\frac{-\left(-283\right)±\sqrt{80089-4\times 15\times 16}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-283\right)±\sqrt{80089-60\times 16}}{2\times 15}$$

-60 ni 16 marotabaga ko'paytirish.

$$x=\frac{-\left(-283\right)±\sqrt{80089-960}}{2\times 15}$$

80089 ni -960 ga qo'shish.

$$x=\frac{-\left(-283\right)±\sqrt{79129}}{2\times 15}$$

-283 ning teskarisi 283 ga teng.

$$x=\frac{283±\sqrt{79129}}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$x=\frac{283±\sqrt{79129}}{30}$$

x=\frac{283±\sqrt{79129}}{30} tenglamasini yeching, bunda ± musbat. 283 ni \sqrt{79129} ga qo'shish.

$$x=\frac{\sqrt{79129}+283}{30}$$

x=\frac{283±\sqrt{79129}}{30} tenglamasini yeching, bunda ± manfiy. 283 dan \sqrt{79129} ni ayirish.

$$x=\frac{283-\sqrt{79129}}{30}$$

Tenglama yechildi.

$$x=\frac{\sqrt{79129}+283}{30} x=\frac{283-\sqrt{79129}}{30}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 19 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 5\left(-x+19\right) ga ko'paytirish.

$$2x+16=3x\times 5\left(-x+19\right)$$

15 hosil qilish uchun 3 va 5 ni ko'paytirish.

$$2x+16=15x\left(-x+19\right)$$

15x ga -x+19 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x+16=-15x^{2}+285x$$

15x^{2} ni ikki tarafga qo’shing.

$$2x+16+15x^{2}=285x$$

Ikkala tarafdan 285x ni ayirish.

$$2x+16+15x^{2}-285x=0$$

-283x ni olish uchun 2x va -285x ni birlashtirish.

$$-283x+16+15x^{2}=0$$

Ikkala tarafdan 16 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-283x+15x^{2}=-16$$

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

$$15x^{2}-283x=-16$$

Ikki tarafini 15 ga bo‘ling.

$$\frac{15x^{2}-283x}{15}=-\frac{16}{15}$$

15 ga bo'lish 15 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{283}{15}x=-\frac{16}{15}$$

-\frac{283}{15} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{283}{30} olish uchun. Keyin, -\frac{283}{30} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{283}{15}x+\left(-\frac{283}{30}\right)^{2}=-\frac{16}{15}+\left(-\frac{283}{30}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{283}{30} kvadratini chiqarish.

$$x^{2}-\frac{283}{15}x+\frac{80089}{900}=-\frac{16}{15}+\frac{80089}{900}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{16}{15} ni \frac{80089}{900} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{283}{15}x+\frac{80089}{900}=\frac{79129}{900}$$

x^{2}-\frac{283}{15}x+\frac{80089}{900} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{283}{30}\right)^{2}=\frac{79129}{900}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{283}{30}\right)^{2}}=\sqrt{\frac{79129}{900}}$$

Qisqartirish.

$$x-\frac{283}{30}=\frac{\sqrt{79129}}{30} x-\frac{283}{30}=-\frac{\sqrt{79129}}{30}$$

\frac{283}{30} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{79129}+283}{30} x=\frac{283-\sqrt{79129}}{30}$$