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

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

x uchun yechish

$$x=-\frac{1}{2}=-0,5$$

$$x=-4$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{1}{2}x ni ikki tarafga qo’shing.

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

-\frac{9}{2}x ni olish uchun -5x va \frac{1}{2}x ni birlashtirish.

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

Ikkala tarafdan 2 ni ayirish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{2} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}+4\left(-2\right)}}{2\left(-1\right)}$$

4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-8}}{2\left(-1\right)}$$

\frac{81}{4} ni -8 ga qo'shish.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{49}{4}}}{2\left(-1\right)}$$

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

$$x=\frac{-\left(-\frac{9}{2}\right)±\frac{7}{2}}{2\left(-1\right)}$$

-\frac{9}{2} ning teskarisi \frac{9}{2} ga teng.

$$x=\frac{\frac{9}{2}±\frac{7}{2}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{\frac{9}{2}±\frac{7}{2}}{-2}$$

x=\frac{\frac{9}{2}±\frac{7}{2}}{-2} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{2} ni \frac{7}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{8}{-2}$$

8 ni -2 ga bo'lish.

$$x=-4$$

x=\frac{\frac{9}{2}±\frac{7}{2}}{-2} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{7}{2} ni \frac{9}{2} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

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

1 ni -2 ga bo'lish.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

\frac{1}{2}x ni ikki tarafga qo’shing.

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

-\frac{9}{2}x ni olish uchun -5x va \frac{1}{2}x ni birlashtirish.

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

Ikki tarafini -1 ga bo‘ling.

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

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

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

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

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

2 ni -1 ga bo'lish.

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

\frac{9}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{9}{4} olish uchun. Keyin, \frac{9}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{4} kvadratini chiqarish.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=-2+\frac{81}{16}$$

-2 ni \frac{81}{16} ga qo'shish.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{49}{16}$$

x^{2}+\frac{9}{2}x+\frac{81}{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.

$$\left(x+\frac{9}{4}\right)^{2}=\frac{49}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{9}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Qisqartirish.

$$x+\frac{9}{4}=\frac{7}{4} x+\frac{9}{4}=-\frac{7}{4}$$

Tenglamaning ikkala tarafidan \frac{9}{4} ni ayirish.

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