15x*(25-x)=2500 eching

$$15 x \times ( 25 - x ) = 2500$$

x uchun yechish (complex solution)

$$x=\frac{5\sqrt{15}i}{6}+\frac{25}{2}\approx 12,5+3,227486122i$$

$$x=-\frac{5\sqrt{15}i}{6}+\frac{25}{2}\approx 12,5-3,227486122i$$

Kvadrat tenglamalaridan foydalanish qoidalari

15x ga 25-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$375x-15x^{2}=2500$$

Ikkala tarafdan 2500 ni ayirish.

$$375x-15x^{2}-2500=0$$

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.

$$-15x^{2}+375x-2500=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -15 ni a, 375 ni b va -2500 ni c bilan almashtiring.

$$x=\frac{-375±\sqrt{375^{2}-4\left(-15\right)\left(-2500\right)}}{2\left(-15\right)}$$

375 kvadratini chiqarish.

$$x=\frac{-375±\sqrt{140625-4\left(-15\right)\left(-2500\right)}}{2\left(-15\right)}$$

-4 ni -15 marotabaga ko'paytirish.

$$x=\frac{-375±\sqrt{140625+60\left(-2500\right)}}{2\left(-15\right)}$$

60 ni -2500 marotabaga ko'paytirish.

$$x=\frac{-375±\sqrt{140625-150000}}{2\left(-15\right)}$$

140625 ni -150000 ga qo'shish.

$$x=\frac{-375±\sqrt{-9375}}{2\left(-15\right)}$$

-9375 ning kvadrat ildizini chiqarish.

$$x=\frac{-375±25\sqrt{15}i}{2\left(-15\right)}$$

2 ni -15 marotabaga ko'paytirish.

$$x=\frac{-375±25\sqrt{15}i}{-30}$$

x=\frac{-375±25\sqrt{15}i}{-30} tenglamasini yeching, bunda ± musbat. -375 ni 25i\sqrt{15} ga qo'shish.

$$x=\frac{-375+25\sqrt{15}i}{-30}$$

-375+25i\sqrt{15} ni -30 ga bo'lish.

$$x=-\frac{5\sqrt{15}i}{6}+\frac{25}{2}$$

x=\frac{-375±25\sqrt{15}i}{-30} tenglamasini yeching, bunda ± manfiy. -375 dan 25i\sqrt{15} ni ayirish.

$$x=\frac{-25\sqrt{15}i-375}{-30}$$

-375-25i\sqrt{15} ni -30 ga bo'lish.

$$x=\frac{5\sqrt{15}i}{6}+\frac{25}{2}$$

Tenglama yechildi.

$$x=-\frac{5\sqrt{15}i}{6}+\frac{25}{2} x=\frac{5\sqrt{15}i}{6}+\frac{25}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

15x ga 25-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$375x-15x^{2}=2500$$

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

$$-15x^{2}+375x=2500$$

Ikki tarafini -15 ga bo‘ling.

$$\frac{-15x^{2}+375x}{-15}=\frac{2500}{-15}$$

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

$$x^{2}+\frac{375}{-15}x=\frac{2500}{-15}$$

375 ni -15 ga bo'lish.

$$x^{2}-25x=\frac{2500}{-15}$$

\frac{2500}{-15} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-25x=-\frac{500}{3}$$

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

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-\frac{500}{3}+\left(-\frac{25}{2}\right)^{2}$$

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

$$x^{2}-25x+\frac{625}{4}=-\frac{500}{3}+\frac{625}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{500}{3} ni \frac{625}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-25x+\frac{625}{4}=-\frac{125}{12}$$

x^{2}-25x+\frac{625}{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.

$$\left(x-\frac{25}{2}\right)^{2}=-\frac{125}{12}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{-\frac{125}{12}}$$

Qisqartirish.

$$x-\frac{25}{2}=\frac{5\sqrt{15}i}{6} x-\frac{25}{2}=-\frac{5\sqrt{15}i}{6}$$

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

$$x=\frac{5\sqrt{15}i}{6}+\frac{25}{2} x=-\frac{5\sqrt{15}i}{6}+\frac{25}{2}$$