left(x+14right)^2-left(x+11right)^2=left(x-6right)^2 eching

$${ \left(x+14 \right) }^{ 2 } - { \left(x+11 \right) }^{ 2 } = { \left(x-6 \right) }^{ 2 }$$

x uchun yechish

$$x=2\sqrt{30}+9\approx 19,95445115$$

$$x=9-2\sqrt{30}\approx -1,95445115$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+14\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+28x+196-\left(x+11\right)^{2}=\left(x-6\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+11\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+28x+196-\left(x^{2}+22x+121\right)=\left(x-6\right)^{2}$$

x^{2}+22x+121 teskarisini topish uchun har birining teskarisini toping.

$$x^{2}+28x+196-x^{2}-22x-121=\left(x-6\right)^{2}$$

0 ni olish uchun x^{2} va -x^{2} ni birlashtirish.

$$28x+196-22x-121=\left(x-6\right)^{2}$$

6x ni olish uchun 28x va -22x ni birlashtirish.

$$6x+196-121=\left(x-6\right)^{2}$$

75 olish uchun 196 dan 121 ni ayirish.

$$6x+75=\left(x-6\right)^{2}$$

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

$$6x+75=x^{2}-12x+36$$

Ikkala tarafdan x^{2} ni ayirish.

$$6x+75-x^{2}=-12x+36$$

12x ni ikki tarafga qo’shing.

$$6x+75-x^{2}+12x=36$$

18x ni olish uchun 6x va 12x ni birlashtirish.

$$18x+75-x^{2}=36$$

Ikkala tarafdan 36 ni ayirish.

$$18x+75-x^{2}-36=0$$

39 olish uchun 75 dan 36 ni ayirish.

$$18x+39-x^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-x^{2}+18x+39=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 18 ni b va 39 ni c bilan almashtiring.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-1\right)\times 39}}{2\left(-1\right)}$$

18 kvadratini chiqarish.

$$x=\frac{-18±\sqrt{324-4\left(-1\right)\times 39}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-18±\sqrt{324+4\times 39}}{2\left(-1\right)}$$

4 ni 39 marotabaga ko'paytirish.

$$x=\frac{-18±\sqrt{324+156}}{2\left(-1\right)}$$

324 ni 156 ga qo'shish.

$$x=\frac{-18±\sqrt{480}}{2\left(-1\right)}$$

480 ning kvadrat ildizini chiqarish.

$$x=\frac{-18±4\sqrt{30}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-18±4\sqrt{30}}{-2}$$

x=\frac{-18±4\sqrt{30}}{-2} tenglamasini yeching, bunda ± musbat. -18 ni 4\sqrt{30} ga qo'shish.

$$x=\frac{4\sqrt{30}-18}{-2}$$

-18+4\sqrt{30} ni -2 ga bo'lish.

$$x=9-2\sqrt{30}$$

x=\frac{-18±4\sqrt{30}}{-2} tenglamasini yeching, bunda ± manfiy. -18 dan 4\sqrt{30} ni ayirish.

$$x=\frac{-4\sqrt{30}-18}{-2}$$

-18-4\sqrt{30} ni -2 ga bo'lish.

$$x=2\sqrt{30}+9$$

Tenglama yechildi.

$$x=9-2\sqrt{30} x=2\sqrt{30}+9$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+14\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+28x+196-\left(x+11\right)^{2}=\left(x-6\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+11\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+28x+196-\left(x^{2}+22x+121\right)=\left(x-6\right)^{2}$$

x^{2}+22x+121 teskarisini topish uchun har birining teskarisini toping.

$$x^{2}+28x+196-x^{2}-22x-121=\left(x-6\right)^{2}$$

0 ni olish uchun x^{2} va -x^{2} ni birlashtirish.

$$28x+196-22x-121=\left(x-6\right)^{2}$$

6x ni olish uchun 28x va -22x ni birlashtirish.

$$6x+196-121=\left(x-6\right)^{2}$$

75 olish uchun 196 dan 121 ni ayirish.

$$6x+75=\left(x-6\right)^{2}$$

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

$$6x+75=x^{2}-12x+36$$

Ikkala tarafdan x^{2} ni ayirish.

$$6x+75-x^{2}=-12x+36$$

12x ni ikki tarafga qo’shing.

$$6x+75-x^{2}+12x=36$$

18x ni olish uchun 6x va 12x ni birlashtirish.

$$18x+75-x^{2}=36$$

Ikkala tarafdan 75 ni ayirish.

$$18x-x^{2}=36-75$$

-39 olish uchun 36 dan 75 ni ayirish.

$$18x-x^{2}=-39$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-x^{2}+18x=-39$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+18x}{-1}=-\frac{39}{-1}$$

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

$$x^{2}+\frac{18}{-1}x=-\frac{39}{-1}$$

18 ni -1 ga bo'lish.

$$x^{2}-18x=-\frac{39}{-1}$$

-39 ni -1 ga bo'lish.

$$x^{2}-18x=39$$

-18 ni bo‘lish, x shartining koeffitsienti, 2 ga -9 olish uchun. Keyin, -9 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-18x+\left(-9\right)^{2}=39+\left(-9\right)^{2}$$

-9 kvadratini chiqarish.

$$x^{2}-18x+81=39+81$$

39 ni 81 ga qo'shish.

$$x^{2}-18x+81=120$$

x^{2}-18x+81 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-9\right)^{2}=120$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{120}$$

Qisqartirish.

$$x-9=2\sqrt{30} x-9=-2\sqrt{30}$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$x=2\sqrt{30}+9 x=9-2\sqrt{30}$$