x^2+(14-x)^2=8^2 eching

$$x ^ { 2 } + ( 14 - x ) ^ { 2 } = 8 ^ { 2 }$$

x uchun yechish (complex solution)

$$x=7+\sqrt{17}i\approx 7+4,123105626i$$

$$x=-\sqrt{17}i+7\approx 7-4,123105626i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x^{2}+196-28x+x^{2}=8^{2}$$

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

$$2x^{2}+196-28x=8^{2}$$

2 daraja ko‘rsatkichini 8 ga hisoblang va 64 ni qiymatni oling.

$$2x^{2}+196-28x=64$$

Ikkala tarafdan 64 ni ayirish.

$$2x^{2}+196-28x-64=0$$

132 olish uchun 196 dan 64 ni ayirish.

$$2x^{2}+132-28x=0$$

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.

$$2x^{2}-28x+132=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -28 ni b va 132 ni c bilan almashtiring.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 2\times 132}}{2\times 2}$$

-28 kvadratini chiqarish.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 2\times 132}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-28\right)±\sqrt{784-8\times 132}}{2\times 2}$$

-8 ni 132 marotabaga ko'paytirish.

$$x=\frac{-\left(-28\right)±\sqrt{784-1056}}{2\times 2}$$

784 ni -1056 ga qo'shish.

$$x=\frac{-\left(-28\right)±\sqrt{-272}}{2\times 2}$$

-272 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-28\right)±4\sqrt{17}i}{2\times 2}$$

-28 ning teskarisi 28 ga teng.

$$x=\frac{28±4\sqrt{17}i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{28±4\sqrt{17}i}{4}$$

x=\frac{28±4\sqrt{17}i}{4} tenglamasini yeching, bunda ± musbat. 28 ni 4i\sqrt{17} ga qo'shish.

$$x=\frac{28+4\sqrt{17}i}{4}$$

28+4i\sqrt{17} ni 4 ga bo'lish.

$$x=7+\sqrt{17}i$$

x=\frac{28±4\sqrt{17}i}{4} tenglamasini yeching, bunda ± manfiy. 28 dan 4i\sqrt{17} ni ayirish.

$$x=\frac{-4\sqrt{17}i+28}{4}$$

28-4i\sqrt{17} ni 4 ga bo'lish.

$$x=-\sqrt{17}i+7$$

Tenglama yechildi.

$$x=7+\sqrt{17}i x=-\sqrt{17}i+7$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$x^{2}+196-28x+x^{2}=8^{2}$$

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

$$2x^{2}+196-28x=8^{2}$$

2 daraja ko‘rsatkichini 8 ga hisoblang va 64 ni qiymatni oling.

$$2x^{2}+196-28x=64$$

Ikkala tarafdan 196 ni ayirish.

$$2x^{2}-28x=64-196$$

-132 olish uchun 64 dan 196 ni ayirish.

$$2x^{2}-28x=-132$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-28x}{2}=-\frac{132}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{28}{2}\right)x=-\frac{132}{2}$$

-28 ni 2 ga bo'lish.

$$x^{2}-14x=-\frac{132}{2}$$

-132 ni 2 ga bo'lish.

$$x^{2}-14x=-66$$

-14 ni bo‘lish, x shartining koeffitsienti, 2 ga -7 olish uchun. Keyin, -7 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-14x+\left(-7\right)^{2}=-66+\left(-7\right)^{2}$$

-7 kvadratini chiqarish.

$$x^{2}-14x+49=-66+49$$

-66 ni 49 ga qo'shish.

$$x^{2}-14x+49=-17$$

x^{2}-14x+49 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-7\right)^{2}=-17$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{-17}$$

Qisqartirish.

$$x-7=\sqrt{17}i x-7=-\sqrt{17}i$$

7 ni tenglamaning ikkala tarafiga qo'shish.

$$x=7+\sqrt{17}i x=-\sqrt{17}i+7$$