-1/2x^2-3/2x+2=2 eching

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

x uchun yechish

$$x=-3$$

$$x=0$$

Faktorlash qadamlari

Ikkala tarafdan 2 ni ayirish.

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

0 olish uchun 2 dan 2 ni ayirish.

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

x omili.

$$x\left(-\frac{1}{2}x-\frac{3}{2}\right)=0$$

Tenglamani yechish uchun x=0 va \frac{-x-3}{2}=0 ni yeching.

$$x=0 x=-3$$

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.

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

Tenglamaning ikkala tarafidan 2 ni ayirish.

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

O‘zidan 2 ayirilsa 0 qoladi.

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

2 dan 2 ni ayirish.

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

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

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

\left(-\frac{3}{2}\right)^{2} ning kvadrat ildizini chiqarish.

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

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

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

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

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

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

$$x=\frac{3}{-1}$$

3 ni -1 ga bo'lish.

$$x=-3$$

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

$$x=\frac{0}{-1}$$

0 ni -1 ga bo'lish.

$$x=0$$

Tenglama yechildi.

$$x=-3 x=0$$

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.

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

Tenglamaning ikkala tarafidan 2 ni ayirish.

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

O‘zidan 2 ayirilsa 0 qoladi.

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

2 dan 2 ni ayirish.

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

Ikkala tarafini -2 ga ko‘paytiring.

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

-\frac{1}{2} ga bo'lish -\frac{1}{2} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{3}{2}}{-\frac{1}{2}}\right)x=\frac{0}{-\frac{1}{2}}$$

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

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

0 ni -\frac{1}{2} ga bo'lish 0 ga k'paytirish -\frac{1}{2} ga qaytarish.

$$x^{2}+3x=0$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\left(\frac{3}{2}\right)^{2}$$

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

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

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

$$\left(x+\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Qisqartirish.

$$x+\frac{3}{2}=\frac{3}{2} x+\frac{3}{2}=-\frac{3}{2}$$

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$x=0 x=-3$$