-h^2+3h+1=4h-1 eching

$$- h ^ { 2 } + 3 h + 1 = 4 h - 1$$

h uchun yechish

$$h=-2$$

$$h=1$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 4h ni ayirish.

$$-h^{2}+3h+1-4h=-1$$

-h ni olish uchun 3h va -4h ni birlashtirish.

$$-h^{2}-h+1=-1$$

1 ni ikki tarafga qo’shing.

$$-h^{2}-h+1+1=0$$

2 olish uchun 1 va 1'ni qo'shing.

$$-h^{2}-h+2=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -h^{2}+ah+bh+2 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-1 ab=-2=-2$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. Faqat bundan juftlik tizim yechimidir.

$$a=1 b=-2$$

-h^{2}-h+2 ni \left(-h^{2}+h\right)+\left(-2h+2\right) sifatida qaytadan yozish.

$$\left(-h^{2}+h\right)+\left(-2h+2\right)$$

Birinchi guruhda h ni va ikkinchi guruhda 2 ni faktordan chiqaring.

$$h\left(-h+1\right)+2\left(-h+1\right)$$

Distributiv funktsiyasidan foydalangan holda -h+1 umumiy terminini chiqaring.

$$\left(-h+1\right)\left(h+2\right)$$

Tenglamani yechish uchun -h+1=0 va h+2=0 ni yeching.

$$h=1 h=-2$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 4h ni ayirish.

$$-h^{2}+3h+1-4h=-1$$

-h ni olish uchun 3h va -4h ni birlashtirish.

$$-h^{2}-h+1=-1$$

1 ni ikki tarafga qo’shing.

$$-h^{2}-h+1+1=0$$

2 olish uchun 1 va 1'ni qo'shing.

$$-h^{2}-h+2=0$$

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

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

-4 ni -1 marotabaga ko'paytirish.

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

4 ni 2 marotabaga ko'paytirish.

$$h=\frac{-\left(-1\right)±\sqrt{1+8}}{2\left(-1\right)}$$

1 ni 8 ga qo'shish.

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

9 ning kvadrat ildizini chiqarish.

$$h=\frac{-\left(-1\right)±3}{2\left(-1\right)}$$

-1 ning teskarisi 1 ga teng.

$$h=\frac{1±3}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$h=\frac{1±3}{-2}$$

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

$$h=\frac{4}{-2}$$

4 ni -2 ga bo'lish.

$$h=-2$$

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

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

-2 ni -2 ga bo'lish.

$$h=1$$

Tenglama yechildi.

$$h=-2 h=1$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 4h ni ayirish.

$$-h^{2}+3h+1-4h=-1$$

-h ni olish uchun 3h va -4h ni birlashtirish.

$$-h^{2}-h+1=-1$$

Ikkala tarafdan 1 ni ayirish.

$$-h^{2}-h=-1-1$$

-2 olish uchun -1 dan 1 ni ayirish.

$$-h^{2}-h=-2$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-h^{2}-h}{-1}=-\frac{2}{-1}$$

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

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

-1 ni -1 ga bo'lish.

$$h^{2}+h=-\frac{2}{-1}$$

-2 ni -1 ga bo'lish.

$$h^{2}+h=2$$

1 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{2} olish uchun. Keyin, \frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$h^{2}+h+\left(\frac{1}{2}\right)^{2}=2+\left(\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$h^{2}+h+\frac{1}{4}=2+\frac{1}{4}$$

2 ni \frac{1}{4} ga qo'shish.

$$h^{2}+h+\frac{1}{4}=\frac{9}{4}$$

h^{2}+h+\frac{1}{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.

$$\left(h+\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(h+\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Qisqartirish.

$$h+\frac{1}{2}=\frac{3}{2} h+\frac{1}{2}=-\frac{3}{2}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$h=1 h=-2$$