(3x-1)^2-(2x+1)^2=7 eching

$$( 3 x - 1 ) ^ { 2 } - ( 2 x + 1 ) ^ { 2 } = 7$$

x uchun yechish

$$x=\frac{2\sqrt{15}}{5}+1\approx 2,549193338$$

$$x=-\frac{2\sqrt{15}}{5}+1\approx -0,549193338$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3x-1\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-6x+1-\left(2x+1\right)^{2}=7$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+1\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-6x+1-\left(4x^{2}+4x+1\right)=7$$

4x^{2}+4x+1 teskarisini topish uchun har birining teskarisini toping.

$$9x^{2}-6x+1-4x^{2}-4x-1=7$$

5x^{2} ni olish uchun 9x^{2} va -4x^{2} ni birlashtirish.

$$5x^{2}-6x+1-4x-1=7$$

-10x ni olish uchun -6x va -4x ni birlashtirish.

$$5x^{2}-10x+1-1=7$$

0 olish uchun 1 dan 1 ni ayirish.

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

Ikkala tarafdan 7 ni ayirish.

$$5x^{2}-10x-7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -10 ni b va -7 ni c bilan almashtiring.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 5\left(-7\right)}}{2\times 5}$$

-10 kvadratini chiqarish.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 5\left(-7\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100-20\left(-7\right)}}{2\times 5}$$

-20 ni -7 marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100+140}}{2\times 5}$$

100 ni 140 ga qo'shish.

$$x=\frac{-\left(-10\right)±\sqrt{240}}{2\times 5}$$

240 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-10\right)±4\sqrt{15}}{2\times 5}$$

-10 ning teskarisi 10 ga teng.

$$x=\frac{10±4\sqrt{15}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{10±4\sqrt{15}}{10}$$

x=\frac{10±4\sqrt{15}}{10} tenglamasini yeching, bunda ± musbat. 10 ni 4\sqrt{15} ga qo'shish.

$$x=\frac{4\sqrt{15}+10}{10}$$

10+4\sqrt{15} ni 10 ga bo'lish.

$$x=\frac{2\sqrt{15}}{5}+1$$

x=\frac{10±4\sqrt{15}}{10} tenglamasini yeching, bunda ± manfiy. 10 dan 4\sqrt{15} ni ayirish.

$$x=\frac{10-4\sqrt{15}}{10}$$

10-4\sqrt{15} ni 10 ga bo'lish.

$$x=-\frac{2\sqrt{15}}{5}+1$$

Tenglama yechildi.

$$x=\frac{2\sqrt{15}}{5}+1 x=-\frac{2\sqrt{15}}{5}+1$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3x-1\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-6x+1-\left(2x+1\right)^{2}=7$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+1\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-6x+1-\left(4x^{2}+4x+1\right)=7$$

4x^{2}+4x+1 teskarisini topish uchun har birining teskarisini toping.

$$9x^{2}-6x+1-4x^{2}-4x-1=7$$

5x^{2} ni olish uchun 9x^{2} va -4x^{2} ni birlashtirish.

$$5x^{2}-6x+1-4x-1=7$$

-10x ni olish uchun -6x va -4x ni birlashtirish.

$$5x^{2}-10x+1-1=7$$

0 olish uchun 1 dan 1 ni ayirish.

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

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}-10x}{5}=\frac{7}{5}$$

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

$$x^{2}+\left(-\frac{10}{5}\right)x=\frac{7}{5}$$

-10 ni 5 ga bo'lish.

$$x^{2}-2x=\frac{7}{5}$$

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

$$x^{2}-2x+1=\frac{7}{5}+1$$

\frac{7}{5} ni 1 ga qo'shish.

$$x^{2}-2x+1=\frac{12}{5}$$

x^{2}-2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-1\right)^{2}=\frac{12}{5}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{12}{5}}$$

Qisqartirish.

$$x-1=\frac{2\sqrt{15}}{5} x-1=-\frac{2\sqrt{15}}{5}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2\sqrt{15}}{5}+1 x=-\frac{2\sqrt{15}}{5}+1$$