-5x^2+3x+4=0 eching

$$-5 { x }^{ 2 } +3x+4 = 0$$

x uchun yechish

$$x = \frac{\sqrt{89} + 3}{10} \approx 1,243398113$$

$$x=\frac{3-\sqrt{89}}{10}\approx -0,643398113$$

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.

$$-5x^{2}+3x+4=0$$

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

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

3 kvadratini chiqarish.

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

-4 ni -5 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+20\times 4}}{2\left(-5\right)}$$

20 ni 4 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+80}}{2\left(-5\right)}$$

9 ni 80 ga qo'shish.

$$x=\frac{-3±\sqrt{89}}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{89}}{-10}$$

x=\frac{-3±\sqrt{89}}{-10} tenglamasini yeching, bunda ± musbat. -3 ni \sqrt{89} ga qo'shish.

$$x=\frac{\sqrt{89}-3}{-10}$$

-3+\sqrt{89} ni -10 ga bo'lish.

$$x=\frac{3-\sqrt{89}}{10}$$

x=\frac{-3±\sqrt{89}}{-10} tenglamasini yeching, bunda ± manfiy. -3 dan \sqrt{89} ni ayirish.

$$x=\frac{-\sqrt{89}-3}{-10}$$

-3-\sqrt{89} ni -10 ga bo'lish.

$$x=\frac{\sqrt{89}+3}{10}$$

Tenglama yechildi.

$$x=\frac{3-\sqrt{89}}{10} x=\frac{\sqrt{89}+3}{10}$$

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.

$$-5x^{2}+3x+4=0$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$-5x^{2}+3x+4-4=-4$$

O‘zidan 4 ayirilsa 0 qoladi.

$$-5x^{2}+3x=-4$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5x^{2}+3x}{-5}=-\frac{4}{-5}$$

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

$$x^{2}+\frac{3}{-5}x=-\frac{4}{-5}$$

3 ni -5 ga bo'lish.

$$x^{2}-\frac{3}{5}x=-\frac{4}{-5}$$

-4 ni -5 ga bo'lish.

$$x^{2}-\frac{3}{5}x=\frac{4}{5}$$

-\frac{3}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{10} olish uchun. Keyin, -\frac{3}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{5}x+\left(-\frac{3}{10}\right)^{2}=\frac{4}{5}+\left(-\frac{3}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{10} kvadratini chiqarish.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=\frac{4}{5}+\frac{9}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{4}{5} ni \frac{9}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=\frac{89}{100}$$

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

$$\left(x-\frac{3}{10}\right)^{2}=\frac{89}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{\frac{89}{100}}$$

Qisqartirish.

$$x-\frac{3}{10}=\frac{\sqrt{89}}{10} x-\frac{3}{10}=-\frac{\sqrt{89}}{10}$$

\frac{3}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{89}+3}{10} x=\frac{3-\sqrt{89}}{10}$$