(4x+3)(7x+5)=2 eching

$$( 4 x + 3 ) ( 7 x + 5 ) = 2$$

x uchun yechish

$$x=-\frac{13}{28}\approx -0,464285714$$

$$x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

4x+3 ga 7x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$28x^{2}+41x+15=2$$

Ikkala tarafdan 2 ni ayirish.

$$28x^{2}+41x+15-2=0$$

13 olish uchun 15 dan 2 ni ayirish.

$$28x^{2}+41x+13=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 28 ni a, 41 ni b va 13 ni c bilan almashtiring.

$$x=\frac{-41±\sqrt{41^{2}-4\times 28\times 13}}{2\times 28}$$

41 kvadratini chiqarish.

$$x=\frac{-41±\sqrt{1681-4\times 28\times 13}}{2\times 28}$$

-4 ni 28 marotabaga ko'paytirish.

$$x=\frac{-41±\sqrt{1681-112\times 13}}{2\times 28}$$

-112 ni 13 marotabaga ko'paytirish.

$$x=\frac{-41±\sqrt{1681-1456}}{2\times 28}$$

1681 ni -1456 ga qo'shish.

$$x=\frac{-41±\sqrt{225}}{2\times 28}$$

225 ning kvadrat ildizini chiqarish.

$$x=\frac{-41±15}{2\times 28}$$

2 ni 28 marotabaga ko'paytirish.

$$x=\frac{-41±15}{56}$$

x=\frac{-41±15}{56} tenglamasini yeching, bunda ± musbat. -41 ni 15 ga qo'shish.

$$x=-\frac{26}{56}$$

\frac{-26}{56} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{13}{28}$$

x=\frac{-41±15}{56} tenglamasini yeching, bunda ± manfiy. -41 dan 15 ni ayirish.

$$x=-\frac{56}{56}$$

-56 ni 56 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

$$x=-\frac{13}{28} x=-1$$

Kvadratni yakunlash bo'yicha qadamlar

4x+3 ga 7x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$28x^{2}+41x+15=2$$

Ikkala tarafdan 15 ni ayirish.

$$28x^{2}+41x=2-15$$

-13 olish uchun 2 dan 15 ni ayirish.

$$28x^{2}+41x=-13$$

Ikki tarafini 28 ga bo‘ling.

$$\frac{28x^{2}+41x}{28}=-\frac{13}{28}$$

28 ga bo'lish 28 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{41}{28}x=-\frac{13}{28}$$

\frac{41}{28} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{41}{56} olish uchun. Keyin, \frac{41}{56} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{41}{28}x+\left(\frac{41}{56}\right)^{2}=-\frac{13}{28}+\left(\frac{41}{56}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{41}{56} kvadratini chiqarish.

$$x^{2}+\frac{41}{28}x+\frac{1681}{3136}=-\frac{13}{28}+\frac{1681}{3136}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{13}{28} ni \frac{1681}{3136} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{41}{28}x+\frac{1681}{3136}=\frac{225}{3136}$$

x^{2}+\frac{41}{28}x+\frac{1681}{3136} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{41}{56}\right)^{2}=\frac{225}{3136}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{41}{56}\right)^{2}}=\sqrt{\frac{225}{3136}}$$

Qisqartirish.

$$x+\frac{41}{56}=\frac{15}{56} x+\frac{41}{56}=-\frac{15}{56}$$

Tenglamaning ikkala tarafidan \frac{41}{56} ni ayirish.

$$x=-\frac{13}{28} x=-1$$