-1/2x^2-3/2x+4=0 eching

$$- \frac { 1 } { 2 } x ^ { 2 } - \frac { 3 } { 2 } x + 4 = 0$$

x uchun yechish

$$x = \frac{\sqrt{41} - 3}{2} \approx 1,701562119$$

$$x=\frac{-\sqrt{41}-3}{2}\approx -4,701562119$$

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.

$$-\frac{1}{2}x^{2}-\frac{3}{2}x+4=0$$

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 4 ni c bilan almashtiring.

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

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

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

-4 ni -\frac{1}{2} marotabaga ko'paytirish.

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

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}+8}}{2\left(-\frac{1}{2}\right)}$$

\frac{9}{4} ni 8 ga qo'shish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{41}{4}}}{2\left(-\frac{1}{2}\right)}$$

\frac{41}{4} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\frac{\sqrt{41}}{2}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{\frac{3}{2}±\frac{\sqrt{41}}{2}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{\frac{3}{2}±\frac{\sqrt{41}}{2}}{-1}$$

x=\frac{\frac{3}{2}±\frac{\sqrt{41}}{2}}{-1} tenglamasini yeching, bunda ± musbat. \frac{3}{2} ni \frac{\sqrt{41}}{2} ga qo'shish.

$$x=\frac{\sqrt{41}+3}{-2}$$

\frac{3+\sqrt{41}}{2} ni -1 ga bo'lish.

$$x=\frac{-\sqrt{41}-3}{2}$$

x=\frac{\frac{3}{2}±\frac{\sqrt{41}}{2}}{-1} tenglamasini yeching, bunda ± manfiy. \frac{3}{2} dan \frac{\sqrt{41}}{2} ni ayirish.

$$x=\frac{3-\sqrt{41}}{-2}$$

\frac{3-\sqrt{41}}{2} ni -1 ga bo'lish.

$$x=\frac{\sqrt{41}-3}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{41}-3}{2} x=\frac{\sqrt{41}-3}{2}$$

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.

$$-\frac{1}{2}x^{2}-\frac{3}{2}x+4=0$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$-\frac{1}{2}x^{2}-\frac{3}{2}x+4-4=-4$$

O‘zidan 4 ayirilsa 0 qoladi.

$$-\frac{1}{2}x^{2}-\frac{3}{2}x=-4$$

Ikkala tarafini -2 ga ko‘paytiring.

$$\frac{-\frac{1}{2}x^{2}-\frac{3}{2}x}{-\frac{1}{2}}=-\frac{4}{-\frac{1}{2}}$$

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

$$x^{2}+\left(-\frac{\frac{3}{2}}{-\frac{1}{2}}\right)x=-\frac{4}{-\frac{1}{2}}$$

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

$$x^{2}+3x=-\frac{4}{-\frac{1}{2}}$$

-4 ni -\frac{1}{2} ga bo'lish -4 ga k'paytirish -\frac{1}{2} ga qaytarish.

$$x^{2}+3x=8$$

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.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=8+\left(\frac{3}{2}\right)^{2}$$

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

$$x^{2}+3x+\frac{9}{4}=8+\frac{9}{4}$$

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

$$x^{2}+3x+\frac{9}{4}=\frac{41}{4}$$

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.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{41}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{41}{4}}$$

Qisqartirish.

$$x+\frac{3}{2}=\frac{\sqrt{41}}{2} x+\frac{3}{2}=-\frac{\sqrt{41}}{2}$$

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

$$x=\frac{\sqrt{41}-3}{2} x=\frac{-\sqrt{41}-3}{2}$$