1/2x+1/2=x^2+6x+5 eching

$$\frac { 1 } { 2 } x + \frac { 1 } { 2 } = x ^ { 2 } + 6 x + 5$$

x uchun yechish

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4,5$$

$$x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan x^{2} ni ayirish.

$$\frac{1}{2}x+\frac{1}{2}-x^{2}=6x+5$$

Ikkala tarafdan 6x ni ayirish.

$$\frac{1}{2}x+\frac{1}{2}-x^{2}-6x=5$$

-\frac{11}{2}x ni olish uchun \frac{1}{2}x va -6x ni birlashtirish.

$$-\frac{11}{2}x+\frac{1}{2}-x^{2}=5$$

Ikkala tarafdan 5 ni ayirish.

$$-\frac{11}{2}x+\frac{1}{2}-x^{2}-5=0$$

-\frac{9}{2} olish uchun \frac{1}{2} dan 5 ni ayirish.

$$-\frac{11}{2}x-\frac{9}{2}-x^{2}=0$$

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.

$$-x^{2}-\frac{11}{2}x-\frac{9}{2}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -\frac{11}{2} ni b va -\frac{9}{2} ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\left(-\frac{11}{2}\right)^{2}-4\left(-1\right)\left(-\frac{9}{2}\right)}}{2\left(-1\right)}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{2} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\frac{121}{4}-4\left(-1\right)\left(-\frac{9}{2}\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\frac{121}{4}+4\left(-\frac{9}{2}\right)}}{2\left(-1\right)}$$

4 ni -\frac{9}{2} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\frac{121}{4}-18}}{2\left(-1\right)}$$

\frac{121}{4} ni -18 ga qo'shish.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\frac{49}{4}}}{2\left(-1\right)}$$

\frac{49}{4} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{11}{2}\right)±\frac{7}{2}}{2\left(-1\right)}$$

-\frac{11}{2} ning teskarisi \frac{11}{2} ga teng.

$$x=\frac{\frac{11}{2}±\frac{7}{2}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{\frac{11}{2}±\frac{7}{2}}{-2}$$

x=\frac{\frac{11}{2}±\frac{7}{2}}{-2} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{11}{2} ni \frac{7}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{9}{-2}$$

9 ni -2 ga bo'lish.

$$x=-\frac{9}{2}$$

x=\frac{\frac{11}{2}±\frac{7}{2}}{-2} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{7}{2} ni \frac{11}{2} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

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

2 ni -2 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

$$x=-\frac{9}{2} x=-1$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan x^{2} ni ayirish.

$$\frac{1}{2}x+\frac{1}{2}-x^{2}=6x+5$$

Ikkala tarafdan 6x ni ayirish.

$$\frac{1}{2}x+\frac{1}{2}-x^{2}-6x=5$$

-\frac{11}{2}x ni olish uchun \frac{1}{2}x va -6x ni birlashtirish.

$$-\frac{11}{2}x+\frac{1}{2}-x^{2}=5$$

Ikkala tarafdan \frac{1}{2} ni ayirish.

$$-\frac{11}{2}x-x^{2}=5-\frac{1}{2}$$

\frac{9}{2} olish uchun 5 dan \frac{1}{2} ni ayirish.

$$-\frac{11}{2}x-x^{2}=\frac{9}{2}$$

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

$$-x^{2}-\frac{11}{2}x=\frac{9}{2}$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-\frac{11}{2}x}{-1}=\frac{\frac{9}{2}}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{11}{2}}{-1}\right)x=\frac{\frac{9}{2}}{-1}$$

-\frac{11}{2} ni -1 ga bo'lish.

$$x^{2}+\frac{11}{2}x=\frac{\frac{9}{2}}{-1}$$

\frac{9}{2} ni -1 ga bo'lish.

$$x^{2}+\frac{11}{2}x=-\frac{9}{2}$$

\frac{11}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{11}{4} olish uchun. Keyin, \frac{11}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{11}{2}x+\left(\frac{11}{4}\right)^{2}=-\frac{9}{2}+\left(\frac{11}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{11}{4} kvadratini chiqarish.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=-\frac{9}{2}+\frac{121}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{9}{2} ni \frac{121}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=\frac{49}{16}$$

x^{2}+\frac{11}{2}x+\frac{121}{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.

$$\left(x+\frac{11}{4}\right)^{2}=\frac{49}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{11}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Qisqartirish.

$$x+\frac{11}{4}=\frac{7}{4} x+\frac{11}{4}=-\frac{7}{4}$$

Tenglamaning ikkala tarafidan \frac{11}{4} ni ayirish.

$$x=-1 x=-\frac{9}{2}$$