-1/5x^2+3x+16/5=0 eching

$$- \frac{ 1 }{ 5 } { x }^{ 2 } +3x+ \frac{ 16 }{ 5 } =0$$

x uchun yechish

$$x=-1$$

$$x=16$$

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.

$$-\frac{1}{5}x^{2}+3x+\frac{16}{5}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -\frac{1}{5} ni a, 3 ni b va \frac{16}{5} ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-\frac{1}{5}\right)\times \frac{16}{5}}}{2\left(-\frac{1}{5}\right)}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\left(-\frac{1}{5}\right)\times \frac{16}{5}}}{2\left(-\frac{1}{5}\right)}$$

-4 ni -\frac{1}{5} marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+\frac{4}{5}\times \frac{16}{5}}}{2\left(-\frac{1}{5}\right)}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{4}{5} ni \frac{16}{5} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=\frac{-3±\sqrt{9+\frac{64}{25}}}{2\left(-\frac{1}{5}\right)}$$

9 ni \frac{64}{25} ga qo'shish.

$$x=\frac{-3±\sqrt{\frac{289}{25}}}{2\left(-\frac{1}{5}\right)}$$

\frac{289}{25} ning kvadrat ildizini chiqarish.

$$x=\frac{-3±\frac{17}{5}}{2\left(-\frac{1}{5}\right)}$$

2 ni -\frac{1}{5} marotabaga ko'paytirish.

$$x=\frac{-3±\frac{17}{5}}{-\frac{2}{5}}$$

x=\frac{-3±\frac{17}{5}}{-\frac{2}{5}} tenglamasini yeching, bunda ± musbat. -3 ni \frac{17}{5} ga qo'shish.

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

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

$$x=-1$$

x=\frac{-3±\frac{17}{5}}{-\frac{2}{5}} tenglamasini yeching, bunda ± manfiy. -3 dan \frac{17}{5} ni ayirish.

$$x=-\frac{\frac{32}{5}}{-\frac{2}{5}}$$

-\frac{32}{5} ni -\frac{2}{5} ga bo'lish -\frac{32}{5} ga k'paytirish -\frac{2}{5} ga qaytarish.

$$x=16$$

Tenglama yechildi.

$$x=-1 x=16$$

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.

$$-\frac{1}{5}x^{2}+3x+\frac{16}{5}=0$$

Tenglamaning ikkala tarafidan \frac{16}{5} ni ayirish.

$$-\frac{1}{5}x^{2}+3x+\frac{16}{5}-\frac{16}{5}=-\frac{16}{5}$$

O‘zidan \frac{16}{5} ayirilsa 0 qoladi.

$$-\frac{1}{5}x^{2}+3x=-\frac{16}{5}$$

Ikkala tarafini -5 ga ko‘paytiring.

$$\frac{-\frac{1}{5}x^{2}+3x}{-\frac{1}{5}}=-\frac{\frac{16}{5}}{-\frac{1}{5}}$$

-\frac{1}{5} ga bo'lish -\frac{1}{5} ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{3}{-\frac{1}{5}}x=-\frac{\frac{16}{5}}{-\frac{1}{5}}$$

3 ni -\frac{1}{5} ga bo'lish 3 ga k'paytirish -\frac{1}{5} ga qaytarish.

$$x^{2}-15x=-\frac{\frac{16}{5}}{-\frac{1}{5}}$$

-\frac{16}{5} ni -\frac{1}{5} ga bo'lish -\frac{16}{5} ga k'paytirish -\frac{1}{5} ga qaytarish.

$$x^{2}-15x=16$$

-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.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=16+\left(-\frac{15}{2}\right)^{2}$$

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

$$x^{2}-15x+\frac{225}{4}=16+\frac{225}{4}$$

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

$$x^{2}-15x+\frac{225}{4}=\frac{289}{4}$$

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.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{289}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Qisqartirish.

$$x-\frac{15}{2}=\frac{17}{2} x-\frac{15}{2}=-\frac{17}{2}$$

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

$$x=16 x=-1$$