1/9x^2+x+9/4=0 eching

$$\frac{ 1 }{ 9 } { x }^{ 2 } +x+ \frac{ 9 }{ 4 } =0$$

x uchun yechish

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4,5$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$\frac{1}{9}x^{2}+x+\frac{9}{4}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{9} ni a, 1 ni b va \frac{9}{4} ni c bilan almashtiring.

$$x=\frac{-1±\sqrt{1^{2}-4\times \frac{1}{9}\times \frac{9}{4}}}{2\times \frac{1}{9}}$$

1 kvadratini chiqarish.

$$x=\frac{-1±\sqrt{1-4\times \frac{1}{9}\times \frac{9}{4}}}{2\times \frac{1}{9}}$$

-4 ni \frac{1}{9} marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1-\frac{4}{9}\times \frac{9}{4}}}{2\times \frac{1}{9}}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali -\frac{4}{9} ni \frac{9}{4} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=\frac{-1±\sqrt{1-1}}{2\times \frac{1}{9}}$$

1 ni -1 ga qo'shish.

$$x=\frac{-1±\sqrt{0}}{2\times \frac{1}{9}}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{1}{2\times \frac{1}{9}}$$

2 ni \frac{1}{9} marotabaga ko'paytirish.

$$x=-\frac{1}{\frac{2}{9}}$$

-1 ni \frac{2}{9} ga bo'lish -1 ga k'paytirish \frac{2}{9} ga qaytarish.

$$x=-\frac{9}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\frac{1}{9}x^{2}+x+\frac{9}{4}=0$$

Tenglamaning ikkala tarafidan \frac{9}{4} ni ayirish.

$$\frac{1}{9}x^{2}+x+\frac{9}{4}-\frac{9}{4}=-\frac{9}{4}$$

O‘zidan \frac{9}{4} ayirilsa 0 qoladi.

$$\frac{1}{9}x^{2}+x=-\frac{9}{4}$$

Ikkala tarafini 9 ga ko‘paytiring.

$$\frac{\frac{1}{9}x^{2}+x}{\frac{1}{9}}=-\frac{\frac{9}{4}}{\frac{1}{9}}$$

\frac{1}{9} ga bo'lish \frac{1}{9} ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{1}{\frac{1}{9}}x=-\frac{\frac{9}{4}}{\frac{1}{9}}$$

1 ni \frac{1}{9} ga bo'lish 1 ga k'paytirish \frac{1}{9} ga qaytarish.

$$x^{2}+9x=-\frac{\frac{9}{4}}{\frac{1}{9}}$$

-\frac{9}{4} ni \frac{1}{9} ga bo'lish -\frac{9}{4} ga k'paytirish \frac{1}{9} ga qaytarish.

$$x^{2}+9x=-\frac{81}{4}$$

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

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-\frac{81}{4}+\left(\frac{9}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{2} kvadratini chiqarish.

$$x^{2}+9x+\frac{81}{4}=\frac{-81+81}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{81}{4} ni \frac{81}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+9x+\frac{81}{4}=0$$

x^{2}+9x+\frac{81}{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.

$$\left(x+\frac{9}{2}\right)^{2}=0$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{0}$$

Qisqartirish.

$$x+\frac{9}{2}=0 x+\frac{9}{2}=0$$

Tenglamaning ikkala tarafidan \frac{9}{2} ni ayirish.

$$x=-\frac{9}{2} x=-\frac{9}{2}$$

Tenglama yechildi. Yechimlar bir xil.

$$x=-\frac{9}{2}$$