(x+3)(6-x)/1-4x^2=-1/4 eching

$$\frac { ( x + 3 ) ( 6 - x ) } { 1 - 4 x ^ { 2 } } = - \frac { 1 } { 4 }$$

x uchun yechish

$$x = \frac{\sqrt{155} + 3}{4} \approx 3,862474899$$

$$x=\frac{3-\sqrt{155}}{4}\approx -2,362474899$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -\frac{1}{2},\frac{1}{2} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(2x-1\right)\left(2x+1\right) ga, 1-4x^{2},4 ning eng kichik karralisiga ko‘paytiring.

$$-4\left(x+3\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

-4 ga x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(-4x-12\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

-4x-12 ga 6-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-12x+4x^{2}-72=-\left(2x-1\right)\left(2x+1\right)$$

-1 ga 2x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-12x+4x^{2}-72=\left(-2x+1\right)\left(2x+1\right)$$

-2x+1 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-12x+4x^{2}-72=-4x^{2}+1$$

4x^{2} ni ikki tarafga qo’shing.

$$-12x+4x^{2}-72+4x^{2}=1$$

8x^{2} ni olish uchun 4x^{2} va 4x^{2} ni birlashtirish.

$$-12x+8x^{2}-72=1$$

Ikkala tarafdan 1 ni ayirish.

$$-12x+8x^{2}-72-1=0$$

-73 olish uchun -72 dan 1 ni ayirish.

$$-12x+8x^{2}-73=0$$

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.

$$8x^{2}-12x-73=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, -12 ni b va -73 ni c bilan almashtiring.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 8\left(-73\right)}}{2\times 8}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 8\left(-73\right)}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-32\left(-73\right)}}{2\times 8}$$

-32 ni -73 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144+2336}}{2\times 8}$$

144 ni 2336 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{2480}}{2\times 8}$$

2480 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±4\sqrt{155}}{2\times 8}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±4\sqrt{155}}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{12±4\sqrt{155}}{16}$$

x=\frac{12±4\sqrt{155}}{16} tenglamasini yeching, bunda ± musbat. 12 ni 4\sqrt{155} ga qo'shish.

$$x=\frac{4\sqrt{155}+12}{16}$$

12+4\sqrt{155} ni 16 ga bo'lish.

$$x=\frac{\sqrt{155}+3}{4}$$

x=\frac{12±4\sqrt{155}}{16} tenglamasini yeching, bunda ± manfiy. 12 dan 4\sqrt{155} ni ayirish.

$$x=\frac{12-4\sqrt{155}}{16}$$

12-4\sqrt{155} ni 16 ga bo'lish.

$$x=\frac{3-\sqrt{155}}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{155}+3}{4} x=\frac{3-\sqrt{155}}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -\frac{1}{2},\frac{1}{2} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(2x-1\right)\left(2x+1\right) ga, 1-4x^{2},4 ning eng kichik karralisiga ko‘paytiring.

$$-4\left(x+3\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

-4 ga x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(-4x-12\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

-4x-12 ga 6-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-12x+4x^{2}-72=-\left(2x-1\right)\left(2x+1\right)$$

-1 ga 2x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-12x+4x^{2}-72=\left(-2x+1\right)\left(2x+1\right)$$

-2x+1 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-12x+4x^{2}-72=-4x^{2}+1$$

4x^{2} ni ikki tarafga qo’shing.

$$-12x+4x^{2}-72+4x^{2}=1$$

8x^{2} ni olish uchun 4x^{2} va 4x^{2} ni birlashtirish.

$$-12x+8x^{2}-72=1$$

72 ni ikki tarafga qo’shing.

$$-12x+8x^{2}=1+72$$

73 olish uchun 1 va 72'ni qo'shing.

$$-12x+8x^{2}=73$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$8x^{2}-12x=73$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}-12x}{8}=\frac{73}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{12}{8}\right)x=\frac{73}{8}$$

\frac{-12}{8} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{3}{2}x=\frac{73}{8}$$

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{73}{8}+\left(-\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{73}{8}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{73}{8} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{155}{16}$$

x^{2}-\frac{3}{2}x+\frac{9}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{155}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{155}{16}}$$

Qisqartirish.

$$x-\frac{3}{4}=\frac{\sqrt{155}}{4} x-\frac{3}{4}=-\frac{\sqrt{155}}{4}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{155}+3}{4} x=\frac{3-\sqrt{155}}{4}$$