1/10x^2-3/2x+5=0 eching

$$\frac { 1 } { 10 } x ^ { 2 } - \frac { 3 } { 2 } x + 5 = 0$$

x uchun yechish

$$x=5$$

$$x=10$$

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.

$$\frac{1}{10}x^{2}-\frac{3}{2}x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{10} ni a, -\frac{3}{2} ni b va 5 ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\left(-\frac{3}{2}\right)^{2}-4\times \frac{1}{10}\times 5}}{2\times \frac{1}{10}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-4\times \frac{1}{10}\times 5}}{2\times \frac{1}{10}}$$

-4 ni \frac{1}{10} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-\frac{2}{5}\times 5}}{2\times \frac{1}{10}}$$

-\frac{2}{5} ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-2}}{2\times \frac{1}{10}}$$

\frac{9}{4} ni -2 ga qo'shish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{1}{4}}}{2\times \frac{1}{10}}$$

\frac{1}{4} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\frac{1}{2}}{2\times \frac{1}{10}}$$

-\frac{3}{2} ning teskarisi \frac{3}{2} ga teng.

$$x=\frac{\frac{3}{2}±\frac{1}{2}}{2\times \frac{1}{10}}$$

2 ni \frac{1}{10} marotabaga ko'paytirish.

$$x=\frac{\frac{3}{2}±\frac{1}{2}}{\frac{1}{5}}$$

x=\frac{\frac{3}{2}±\frac{1}{2}}{\frac{1}{5}} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{2} ni \frac{1}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{2}{\frac{1}{5}}$$

2 ni \frac{1}{5} ga bo'lish 2 ga k'paytirish \frac{1}{5} ga qaytarish.

$$x=10$$

x=\frac{\frac{3}{2}±\frac{1}{2}}{\frac{1}{5}} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{1}{2} ni \frac{3}{2} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$x=\frac{1}{\frac{1}{5}}$$

1 ni \frac{1}{5} ga bo'lish 1 ga k'paytirish \frac{1}{5} ga qaytarish.

$$x=5$$

Tenglama yechildi.

$$x=10 x=5$$

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.

$$\frac{1}{10}x^{2}-\frac{3}{2}x+5=0$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$\frac{1}{10}x^{2}-\frac{3}{2}x+5-5=-5$$

O‘zidan 5 ayirilsa 0 qoladi.

$$\frac{1}{10}x^{2}-\frac{3}{2}x=-5$$

Ikkala tarafini 10 ga ko‘paytiring.

$$\frac{\frac{1}{10}x^{2}-\frac{3}{2}x}{\frac{1}{10}}=-\frac{5}{\frac{1}{10}}$$

\frac{1}{10} ga bo'lish \frac{1}{10} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{3}{2}}{\frac{1}{10}}\right)x=-\frac{5}{\frac{1}{10}}$$

-\frac{3}{2} ni \frac{1}{10} ga bo'lish -\frac{3}{2} ga k'paytirish \frac{1}{10} ga qaytarish.

$$x^{2}-15x=-\frac{5}{\frac{1}{10}}$$

-5 ni \frac{1}{10} ga bo'lish -5 ga k'paytirish \frac{1}{10} ga qaytarish.

$$x^{2}-15x=-50$$

-15 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{2} olish uchun. Keyin, -\frac{15}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-50+\left(-\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{2} kvadratini chiqarish.

$$x^{2}-15x+\frac{225}{4}=-50+\frac{225}{4}$$

-50 ni \frac{225}{4} ga qo'shish.

$$x^{2}-15x+\frac{225}{4}=\frac{25}{4}$$

x^{2}-15x+\frac{225}{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.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$x-\frac{15}{2}=\frac{5}{2} x-\frac{15}{2}=-\frac{5}{2}$$

\frac{15}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=10 x=5$$