(-2x+9)(-9x+5)+(-9x-5)^2=0 eching

$$( - 2 x + 9 ) ( - 9 x + 5 ) + ( - 9 x - 5 ) ^ { 2 } = 0$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{27719}i}{198}\approx 0,005050505+0,840859798i$$

$$x=\frac{-\sqrt{27719}i+1}{198}\approx 0,005050505-0,840859798i$$

Kvadrat tenglamalaridan foydalanish qoidalari

-2x+9 ga -9x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$18x^{2}-91x+45+\left(-9x-5\right)^{2}=0$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-9x-5\right)^{2} kengaytirilishi uchun ishlating.

$$18x^{2}-91x+45+81x^{2}+90x+25=0$$

99x^{2} ni olish uchun 18x^{2} va 81x^{2} ni birlashtirish.

$$99x^{2}-91x+45+90x+25=0$$

-x ni olish uchun -91x va 90x ni birlashtirish.

$$99x^{2}-x+45+25=0$$

70 olish uchun 45 va 25'ni qo'shing.

$$99x^{2}-x+70=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 99 ni a, -1 ni b va 70 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 99\times 70}}{2\times 99}$$

-4 ni 99 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-396\times 70}}{2\times 99}$$

-396 ni 70 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-27720}}{2\times 99}$$

1 ni -27720 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{-27719}}{2\times 99}$$

-27719 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±\sqrt{27719}i}{2\times 99}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{27719}i}{2\times 99}$$

2 ni 99 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{27719}i}{198}$$

x=\frac{1±\sqrt{27719}i}{198} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{27719} ga qo'shish.

$$x=\frac{1+\sqrt{27719}i}{198}$$

x=\frac{1±\sqrt{27719}i}{198} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{27719} ni ayirish.

$$x=\frac{-\sqrt{27719}i+1}{198}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{27719}i}{198} x=\frac{-\sqrt{27719}i+1}{198}$$

Kvadratni yakunlash bo'yicha qadamlar

-2x+9 ga -9x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$18x^{2}-91x+45+\left(-9x-5\right)^{2}=0$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-9x-5\right)^{2} kengaytirilishi uchun ishlating.

$$18x^{2}-91x+45+81x^{2}+90x+25=0$$

99x^{2} ni olish uchun 18x^{2} va 81x^{2} ni birlashtirish.

$$99x^{2}-91x+45+90x+25=0$$

-x ni olish uchun -91x va 90x ni birlashtirish.

$$99x^{2}-x+45+25=0$$

70 olish uchun 45 va 25'ni qo'shing.

$$99x^{2}-x+70=0$$

Ikkala tarafdan 70 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$99x^{2}-x=-70$$

Ikki tarafini 99 ga bo‘ling.

$$\frac{99x^{2}-x}{99}=-\frac{70}{99}$$

99 ga bo'lish 99 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{1}{99}x=-\frac{70}{99}$$

-\frac{1}{99} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{198} olish uchun. Keyin, -\frac{1}{198} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{99}x+\left(-\frac{1}{198}\right)^{2}=-\frac{70}{99}+\left(-\frac{1}{198}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{198} kvadratini chiqarish.

$$x^{2}-\frac{1}{99}x+\frac{1}{39204}=-\frac{70}{99}+\frac{1}{39204}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{70}{99} ni \frac{1}{39204} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{99}x+\frac{1}{39204}=-\frac{27719}{39204}$$

x^{2}-\frac{1}{99}x+\frac{1}{39204} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{198}\right)^{2}=-\frac{27719}{39204}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{198}\right)^{2}}=\sqrt{-\frac{27719}{39204}}$$

Qisqartirish.

$$x-\frac{1}{198}=\frac{\sqrt{27719}i}{198} x-\frac{1}{198}=-\frac{\sqrt{27719}i}{198}$$

\frac{1}{198} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{27719}i}{198} x=\frac{-\sqrt{27719}i+1}{198}$$