(4x+6)^2=2x+3 eching

$$( 4 x + 6 ) ^ { 2 } = 2 x + 3$$

x uchun yechish

$$x = -\frac{11}{8} = -1\frac{3}{8} = -1,375$$

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1,5$$

Guruhlash orqali faktorlash qadamlari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(4x+6\right)^{2} kengaytirilishi uchun ishlating.

$$16x^{2}+48x+36=2x+3$$

Ikkala tarafdan 2x ni ayirish.

$$16x^{2}+48x+36-2x=3$$

46x ni olish uchun 48x va -2x ni birlashtirish.

$$16x^{2}+46x+36=3$$

Ikkala tarafdan 3 ni ayirish.

$$16x^{2}+46x+36-3=0$$

33 olish uchun 36 dan 3 ni ayirish.

$$16x^{2}+46x+33=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 16x^{2}+ax+bx+33 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=46 ab=16\times 33=528$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 528-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,528 2,264 3,176 4,132 6,88 8,66 11,48 12,44 16,33 22,24$$

Har bir juftlik yigʻindisini hisoblang.

$$1+528=529 2+264=266 3+176=179 4+132=136 6+88=94 8+66=74 11+48=59 12+44=56 16+33=49 22+24=46$$

Yechim – 46 yigʻindisini beruvchi juftlik.

$$a=22 b=24$$

16x^{2}+46x+33 ni \left(16x^{2}+22x\right)+\left(24x+33\right) sifatida qaytadan yozish.

$$\left(16x^{2}+22x\right)+\left(24x+33\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$2x\left(8x+11\right)+3\left(8x+11\right)$$

Distributiv funktsiyasidan foydalangan holda 8x+11 umumiy terminini chiqaring.

$$\left(8x+11\right)\left(2x+3\right)$$

Tenglamani yechish uchun 8x+11=0 va 2x+3=0 ni yeching.

$$x=-\frac{11}{8} x=-\frac{3}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(4x+6\right)^{2} kengaytirilishi uchun ishlating.

$$16x^{2}+48x+36=2x+3$$

Ikkala tarafdan 2x ni ayirish.

$$16x^{2}+48x+36-2x=3$$

46x ni olish uchun 48x va -2x ni birlashtirish.

$$16x^{2}+46x+36=3$$

Ikkala tarafdan 3 ni ayirish.

$$16x^{2}+46x+36-3=0$$

33 olish uchun 36 dan 3 ni ayirish.

$$16x^{2}+46x+33=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 16 ni a, 46 ni b va 33 ni c bilan almashtiring.

$$x=\frac{-46±\sqrt{46^{2}-4\times 16\times 33}}{2\times 16}$$

46 kvadratini chiqarish.

$$x=\frac{-46±\sqrt{2116-4\times 16\times 33}}{2\times 16}$$

-4 ni 16 marotabaga ko'paytirish.

$$x=\frac{-46±\sqrt{2116-64\times 33}}{2\times 16}$$

-64 ni 33 marotabaga ko'paytirish.

$$x=\frac{-46±\sqrt{2116-2112}}{2\times 16}$$

2116 ni -2112 ga qo'shish.

$$x=\frac{-46±\sqrt{4}}{2\times 16}$$

4 ning kvadrat ildizini chiqarish.

$$x=\frac{-46±2}{2\times 16}$$

2 ni 16 marotabaga ko'paytirish.

$$x=\frac{-46±2}{32}$$

x=\frac{-46±2}{32} tenglamasini yeching, bunda ± musbat. -46 ni 2 ga qo'shish.

$$x=-\frac{44}{32}$$

\frac{-44}{32} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{11}{8}$$

x=\frac{-46±2}{32} tenglamasini yeching, bunda ± manfiy. -46 dan 2 ni ayirish.

$$x=-\frac{48}{32}$$

\frac{-48}{32} ulushini 16 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{3}{2}$$

Tenglama yechildi.

$$x=-\frac{11}{8} x=-\frac{3}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(4x+6\right)^{2} kengaytirilishi uchun ishlating.

$$16x^{2}+48x+36=2x+3$$

Ikkala tarafdan 2x ni ayirish.

$$16x^{2}+48x+36-2x=3$$

46x ni olish uchun 48x va -2x ni birlashtirish.

$$16x^{2}+46x+36=3$$

Ikkala tarafdan 36 ni ayirish.

$$16x^{2}+46x=3-36$$

-33 olish uchun 3 dan 36 ni ayirish.

$$16x^{2}+46x=-33$$

Ikki tarafini 16 ga bo‘ling.

$$\frac{16x^{2}+46x}{16}=-\frac{33}{16}$$

16 ga bo'lish 16 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{46}{16}x=-\frac{33}{16}$$

\frac{46}{16} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{23}{8}x=-\frac{33}{16}$$

\frac{23}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{23}{16} olish uchun. Keyin, \frac{23}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{23}{8}x+\left(\frac{23}{16}\right)^{2}=-\frac{33}{16}+\left(\frac{23}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{23}{16} kvadratini chiqarish.

$$x^{2}+\frac{23}{8}x+\frac{529}{256}=-\frac{33}{16}+\frac{529}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{33}{16} ni \frac{529}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{23}{8}x+\frac{529}{256}=\frac{1}{256}$$

x^{2}+\frac{23}{8}x+\frac{529}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{23}{16}\right)^{2}=\frac{1}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{23}{16}\right)^{2}}=\sqrt{\frac{1}{256}}$$

Qisqartirish.

$$x+\frac{23}{16}=\frac{1}{16} x+\frac{23}{16}=-\frac{1}{16}$$

Tenglamaning ikkala tarafidan \frac{23}{16} ni ayirish.

$$x=-\frac{11}{8} x=-\frac{3}{2}$$