-39+(2x-3)^2=2(-10) eching

$$- 39 + ( 2 x - 3 ) ^ { 2 } = 2 ( - 10 )$$

x uchun yechish

$$x = \frac{\sqrt{19} + 3}{2} \approx 3,679449472$$

$$x=\frac{3-\sqrt{19}}{2}\approx -0,679449472$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-3\right)^{2} kengaytirilishi uchun ishlating.

$$-39+4x^{2}-12x+9=2\left(-10\right)$$

-30 olish uchun -39 va 9'ni qo'shing.

$$-30+4x^{2}-12x=2\left(-10\right)$$

-20 hosil qilish uchun 2 va -10 ni ko'paytirish.

$$-30+4x^{2}-12x=-20$$

20 ni ikki tarafga qo’shing.

$$-30+4x^{2}-12x+20=0$$

-10 olish uchun -30 va 20'ni qo'shing.

$$-10+4x^{2}-12x=0$$

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.

$$4x^{2}-12x-10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -12 ni b va -10 ni c bilan almashtiring.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-10\right)}}{2\times 4}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-10\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\left(-10\right)}}{2\times 4}$$

-16 ni -10 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144+160}}{2\times 4}$$

144 ni 160 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{304}}{2\times 4}$$

304 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±4\sqrt{19}}{2\times 4}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±4\sqrt{19}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{12±4\sqrt{19}}{8}$$

x=\frac{12±4\sqrt{19}}{8} tenglamasini yeching, bunda ± musbat. 12 ni 4\sqrt{19} ga qo'shish.

$$x=\frac{4\sqrt{19}+12}{8}$$

12+4\sqrt{19} ni 8 ga bo'lish.

$$x=\frac{\sqrt{19}+3}{2}$$

x=\frac{12±4\sqrt{19}}{8} tenglamasini yeching, bunda ± manfiy. 12 dan 4\sqrt{19} ni ayirish.

$$x=\frac{12-4\sqrt{19}}{8}$$

12-4\sqrt{19} ni 8 ga bo'lish.

$$x=\frac{3-\sqrt{19}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{19}+3}{2} x=\frac{3-\sqrt{19}}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-3\right)^{2} kengaytirilishi uchun ishlating.

$$-39+4x^{2}-12x+9=2\left(-10\right)$$

-30 olish uchun -39 va 9'ni qo'shing.

$$-30+4x^{2}-12x=2\left(-10\right)$$

-20 hosil qilish uchun 2 va -10 ni ko'paytirish.

$$-30+4x^{2}-12x=-20$$

30 ni ikki tarafga qo’shing.

$$4x^{2}-12x=-20+30$$

10 olish uchun -20 va 30'ni qo'shing.

$$4x^{2}-12x=10$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-12x}{4}=\frac{10}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{12}{4}\right)x=\frac{10}{4}$$

-12 ni 4 ga bo'lish.

$$x^{2}-3x=\frac{10}{4}$$

\frac{10}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

-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.

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

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

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

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni \frac{9}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-3x+\frac{9}{4}=\frac{19}{4}$$

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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{19}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{19}{4}}$$

Qisqartirish.

$$x-\frac{3}{2}=\frac{\sqrt{19}}{2} x-\frac{3}{2}=-\frac{\sqrt{19}}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{19}+3}{2} x=\frac{3-\sqrt{19}}{2}$$