-5x^2+9x=-3 eching

$$- 5 x ^ { 2 } + 9 x = - 3$$

x uchun yechish

$$x = \frac{\sqrt{141} + 9}{10} \approx 2,087434209$$

$$x=\frac{9-\sqrt{141}}{10}\approx -0,287434209$$

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.

$$-5x^{2}+9x=-3$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$-5x^{2}+9x-\left(-3\right)=-3-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$-5x^{2}+9x-\left(-3\right)=0$$

0 dan -3 ni ayirish.

$$-5x^{2}+9x+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, 9 ni b va 3 ni c bilan almashtiring.

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

9 kvadratini chiqarish.

$$x=\frac{-9±\sqrt{81-4\left(-5\right)\times 3}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+20\times 3}}{2\left(-5\right)}$$

20 ni 3 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+60}}{2\left(-5\right)}$$

81 ni 60 ga qo'shish.

$$x=\frac{-9±\sqrt{141}}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{141}}{-10}$$

x=\frac{-9±\sqrt{141}}{-10} tenglamasini yeching, bunda ± musbat. -9 ni \sqrt{141} ga qo'shish.

$$x=\frac{\sqrt{141}-9}{-10}$$

-9+\sqrt{141} ni -10 ga bo'lish.

$$x=\frac{9-\sqrt{141}}{10}$$

x=\frac{-9±\sqrt{141}}{-10} tenglamasini yeching, bunda ± manfiy. -9 dan \sqrt{141} ni ayirish.

$$x=\frac{-\sqrt{141}-9}{-10}$$

-9-\sqrt{141} ni -10 ga bo'lish.

$$x=\frac{\sqrt{141}+9}{10}$$

Tenglama yechildi.

$$x=\frac{9-\sqrt{141}}{10} x=\frac{\sqrt{141}+9}{10}$$

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.

$$-5x^{2}+9x=-3$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5x^{2}+9x}{-5}=-\frac{3}{-5}$$

-5 ga bo'lish -5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{9}{-5}x=-\frac{3}{-5}$$

9 ni -5 ga bo'lish.

$$x^{2}-\frac{9}{5}x=-\frac{3}{-5}$$

-3 ni -5 ga bo'lish.

$$x^{2}-\frac{9}{5}x=\frac{3}{5}$$

-\frac{9}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{10} olish uchun. Keyin, -\frac{9}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{9}{5}x+\left(-\frac{9}{10}\right)^{2}=\frac{3}{5}+\left(-\frac{9}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{10} kvadratini chiqarish.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=\frac{3}{5}+\frac{81}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{5} ni \frac{81}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=\frac{141}{100}$$

x^{2}-\frac{9}{5}x+\frac{81}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{9}{10}\right)^{2}=\frac{141}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{10}\right)^{2}}=\sqrt{\frac{141}{100}}$$

Qisqartirish.

$$x-\frac{9}{10}=\frac{\sqrt{141}}{10} x-\frac{9}{10}=-\frac{\sqrt{141}}{10}$$

\frac{9}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{141}+9}{10} x=\frac{9-\sqrt{141}}{10}$$