90000=120-625(x-9)^2 eching

$$90000 = 120 - 625 ( x - 9 ) ^ { 2 }$$

x uchun yechish (complex solution)

$$x=-\frac{2\sqrt{22470}i}{25}+9\approx 9-11,991997332i$$

$$x=\frac{2\sqrt{22470}i}{25}+9\approx 9+11,991997332i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-9\right)^{2} kengaytirilishi uchun ishlating.

$$90000=120-625\left(x^{2}-18x+81\right)$$

-625 ga x^{2}-18x+81 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$90000=120-625x^{2}+11250x-50625$$

-50505 olish uchun 120 dan 50625 ni ayirish.

$$90000=-50505-625x^{2}+11250x$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-50505-625x^{2}+11250x=90000$$

Ikkala tarafdan 90000 ni ayirish.

$$-50505-625x^{2}+11250x-90000=0$$

-140505 olish uchun -50505 dan 90000 ni ayirish.

$$-140505-625x^{2}+11250x=0$$

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.

$$-625x^{2}+11250x-140505=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -625 ni a, 11250 ni b va -140505 ni c bilan almashtiring.

$$x=\frac{-11250±\sqrt{11250^{2}-4\left(-625\right)\left(-140505\right)}}{2\left(-625\right)}$$

11250 kvadratini chiqarish.

$$x=\frac{-11250±\sqrt{126562500-4\left(-625\right)\left(-140505\right)}}{2\left(-625\right)}$$

-4 ni -625 marotabaga ko'paytirish.

$$x=\frac{-11250±\sqrt{126562500+2500\left(-140505\right)}}{2\left(-625\right)}$$

2500 ni -140505 marotabaga ko'paytirish.

$$x=\frac{-11250±\sqrt{126562500-351262500}}{2\left(-625\right)}$$

126562500 ni -351262500 ga qo'shish.

$$x=\frac{-11250±\sqrt{-224700000}}{2\left(-625\right)}$$

-224700000 ning kvadrat ildizini chiqarish.

$$x=\frac{-11250±100\sqrt{22470}i}{2\left(-625\right)}$$

2 ni -625 marotabaga ko'paytirish.

$$x=\frac{-11250±100\sqrt{22470}i}{-1250}$$

x=\frac{-11250±100\sqrt{22470}i}{-1250} tenglamasini yeching, bunda ± musbat. -11250 ni 100i\sqrt{22470} ga qo'shish.

$$x=\frac{-11250+100\sqrt{22470}i}{-1250}$$

-11250+100i\sqrt{22470} ni -1250 ga bo'lish.

$$x=-\frac{2\sqrt{22470}i}{25}+9$$

x=\frac{-11250±100\sqrt{22470}i}{-1250} tenglamasini yeching, bunda ± manfiy. -11250 dan 100i\sqrt{22470} ni ayirish.

$$x=\frac{-100\sqrt{22470}i-11250}{-1250}$$

-11250-100i\sqrt{22470} ni -1250 ga bo'lish.

$$x=\frac{2\sqrt{22470}i}{25}+9$$

Tenglama yechildi.

$$x=-\frac{2\sqrt{22470}i}{25}+9 x=\frac{2\sqrt{22470}i}{25}+9$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-9\right)^{2} kengaytirilishi uchun ishlating.

$$90000=120-625\left(x^{2}-18x+81\right)$$

-625 ga x^{2}-18x+81 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$90000=120-625x^{2}+11250x-50625$$

-50505 olish uchun 120 dan 50625 ni ayirish.

$$90000=-50505-625x^{2}+11250x$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-50505-625x^{2}+11250x=90000$$

50505 ni ikki tarafga qo’shing.

$$-625x^{2}+11250x=90000+50505$$

140505 olish uchun 90000 va 50505'ni qo'shing.

$$-625x^{2}+11250x=140505$$

Ikki tarafini -625 ga bo‘ling.

$$\frac{-625x^{2}+11250x}{-625}=\frac{140505}{-625}$$

-625 ga bo'lish -625 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{11250}{-625}x=\frac{140505}{-625}$$

11250 ni -625 ga bo'lish.

$$x^{2}-18x=\frac{140505}{-625}$$

\frac{140505}{-625} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-18x=-\frac{28101}{125}$$

-18 ni bo‘lish, x shartining koeffitsienti, 2 ga -9 olish uchun. Keyin, -9 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-18x+\left(-9\right)^{2}=-\frac{28101}{125}+\left(-9\right)^{2}$$

-9 kvadratini chiqarish.

$$x^{2}-18x+81=-\frac{28101}{125}+81$$

-\frac{28101}{125} ni 81 ga qo'shish.

$$x^{2}-18x+81=-\frac{17976}{125}$$

x^{2}-18x+81 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-9\right)^{2}=-\frac{17976}{125}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{-\frac{17976}{125}}$$

Qisqartirish.

$$x-9=\frac{2\sqrt{22470}i}{25} x-9=-\frac{2\sqrt{22470}i}{25}$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2\sqrt{22470}i}{25}+9 x=-\frac{2\sqrt{22470}i}{25}+9$$