5x^2-7x-6=-10x-4 eching

$$5 x ^ { 2 } - 7 x - 6 = - 10 x - 4$$

x uchun yechish

$$x=-1$$

$$x=\frac{2}{5}=0,4$$

Guruhlash orqali faktorlash qadamlari

10x ni ikki tarafga qo’shing.

$$5x^{2}-7x-6+10x=-4$$

3x ni olish uchun -7x va 10x ni birlashtirish.

$$5x^{2}+3x-6=-4$$

4 ni ikki tarafga qo’shing.

$$5x^{2}+3x-6+4=0$$

-2 olish uchun -6 va 4'ni qo'shing.

$$5x^{2}+3x-2=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 5x^{2}+ax+bx-2 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=3 ab=5\left(-2\right)=-10$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -10-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,10 -2,5$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+10=9 -2+5=3$$

Yechim – 3 yigʻindisini beruvchi juftlik.

$$a=-2 b=5$$

5x^{2}+3x-2 ni \left(5x^{2}-2x\right)+\left(5x-2\right) sifatida qaytadan yozish.

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

5x^{2}-2x ichida x ni ajrating.

$$x\left(5x-2\right)+5x-2$$

Distributiv funktsiyasidan foydalangan holda 5x-2 umumiy terminini chiqaring.

$$\left(5x-2\right)\left(x+1\right)$$

Tenglamani yechish uchun 5x-2=0 va x+1=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

10x ni ikki tarafga qo’shing.

$$5x^{2}-7x-6+10x=-4$$

3x ni olish uchun -7x va 10x ni birlashtirish.

$$5x^{2}+3x-6=-4$$

4 ni ikki tarafga qo’shing.

$$5x^{2}+3x-6+4=0$$

-2 olish uchun -6 va 4'ni qo'shing.

$$5x^{2}+3x-2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 3 ni b va -2 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\times 5\left(-2\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9-20\left(-2\right)}}{2\times 5}$$

-20 ni -2 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+40}}{2\times 5}$$

9 ni 40 ga qo'shish.

$$x=\frac{-3±\sqrt{49}}{2\times 5}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±7}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-3±7}{10}$$

x=\frac{-3±7}{10} tenglamasini yeching, bunda ± musbat. -3 ni 7 ga qo'shish.

$$x=\frac{4}{10}$$

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

$$x=\frac{2}{5}$$

x=\frac{-3±7}{10} tenglamasini yeching, bunda ± manfiy. -3 dan 7 ni ayirish.

$$x=-\frac{10}{10}$$

-10 ni 10 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

10x ni ikki tarafga qo’shing.

$$5x^{2}-7x-6+10x=-4$$

3x ni olish uchun -7x va 10x ni birlashtirish.

$$5x^{2}+3x-6=-4$$

6 ni ikki tarafga qo’shing.

$$5x^{2}+3x=-4+6$$

2 olish uchun -4 va 6'ni qo'shing.

$$5x^{2}+3x=2$$

Ikki tarafini 5 ga bo‘ling.

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

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

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

\frac{3}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{10} olish uchun. Keyin, \frac{3}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{3}{5}x+\left(\frac{3}{10}\right)^{2}=\frac{2}{5}+\left(\frac{3}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{10} kvadratini chiqarish.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=\frac{2}{5}+\frac{9}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{5} ni \frac{9}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=\frac{49}{100}$$

x^{2}+\frac{3}{5}x+\frac{9}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{3}{10}\right)^{2}=\frac{49}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{10}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Qisqartirish.

$$x+\frac{3}{10}=\frac{7}{10} x+\frac{3}{10}=-\frac{7}{10}$$

Tenglamaning ikkala tarafidan \frac{3}{10} ni ayirish.

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