x=-16x^2+81x+5 eching

$$x=-16 { x }^{ 2 } +81x+5$$

x uchun yechish

$$x=\frac{\sqrt{105}}{4}+\frac{5}{2}\approx 5,061737691$$

$$x=-\frac{\sqrt{105}}{4}+\frac{5}{2}\approx -0,061737691$$

Kvadrat tenglamalaridan foydalanish qoidalari

16x^{2} ni ikki tarafga qo’shing.

$$x+16x^{2}=81x+5$$

Ikkala tarafdan 81x ni ayirish.

$$x+16x^{2}-81x=5$$

-80x ni olish uchun x va -81x ni birlashtirish.

$$-80x+16x^{2}=5$$

Ikkala tarafdan 5 ni ayirish.

$$-80x+16x^{2}-5=0$$

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.

$$16x^{2}-80x-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 16 ni a, -80 ni b va -5 ni c bilan almashtiring.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 16\left(-5\right)}}{2\times 16}$$

-80 kvadratini chiqarish.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 16\left(-5\right)}}{2\times 16}$$

-4 ni 16 marotabaga ko'paytirish.

$$x=\frac{-\left(-80\right)±\sqrt{6400-64\left(-5\right)}}{2\times 16}$$

-64 ni -5 marotabaga ko'paytirish.

$$x=\frac{-\left(-80\right)±\sqrt{6400+320}}{2\times 16}$$

6400 ni 320 ga qo'shish.

$$x=\frac{-\left(-80\right)±\sqrt{6720}}{2\times 16}$$

6720 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-80\right)±8\sqrt{105}}{2\times 16}$$

-80 ning teskarisi 80 ga teng.

$$x=\frac{80±8\sqrt{105}}{2\times 16}$$

2 ni 16 marotabaga ko'paytirish.

$$x=\frac{80±8\sqrt{105}}{32}$$

x=\frac{80±8\sqrt{105}}{32} tenglamasini yeching, bunda ± musbat. 80 ni 8\sqrt{105} ga qo'shish.

$$x=\frac{8\sqrt{105}+80}{32}$$

80+8\sqrt{105} ni 32 ga bo'lish.

$$x=\frac{\sqrt{105}}{4}+\frac{5}{2}$$

x=\frac{80±8\sqrt{105}}{32} tenglamasini yeching, bunda ± manfiy. 80 dan 8\sqrt{105} ni ayirish.

$$x=\frac{80-8\sqrt{105}}{32}$$

80-8\sqrt{105} ni 32 ga bo'lish.

$$x=-\frac{\sqrt{105}}{4}+\frac{5}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{105}}{4}+\frac{5}{2} x=-\frac{\sqrt{105}}{4}+\frac{5}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

16x^{2} ni ikki tarafga qo’shing.

$$x+16x^{2}=81x+5$$

Ikkala tarafdan 81x ni ayirish.

$$x+16x^{2}-81x=5$$

-80x ni olish uchun x va -81x ni birlashtirish.

$$-80x+16x^{2}=5$$

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

$$16x^{2}-80x=5$$

Ikki tarafini 16 ga bo‘ling.

$$\frac{16x^{2}-80x}{16}=\frac{5}{16}$$

16 ga bo'lish 16 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{80}{16}\right)x=\frac{5}{16}$$

-80 ni 16 ga bo'lish.

$$x^{2}-5x=\frac{5}{16}$$

-5 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{2} olish uchun. Keyin, -\frac{5}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=\frac{5}{16}+\left(-\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{2} kvadratini chiqarish.

$$x^{2}-5x+\frac{25}{4}=\frac{5}{16}+\frac{25}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{16} ni \frac{25}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-5x+\frac{25}{4}=\frac{105}{16}$$

x^{2}-5x+\frac{25}{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.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{105}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{105}{16}}$$

Qisqartirish.

$$x-\frac{5}{2}=\frac{\sqrt{105}}{4} x-\frac{5}{2}=-\frac{\sqrt{105}}{4}$$

\frac{5}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{105}}{4}+\frac{5}{2} x=-\frac{\sqrt{105}}{4}+\frac{5}{2}$$