(sqrt{x+16})^2=(x+1)^2 eching

$$( \sqrt { x + 16 } ) ^ { 2 } = ( x + 1 ) ^ { 2 }$$

x uchun yechish

$$x = \frac{\sqrt{61} - 1}{2} \approx 3,405124838$$

$$x=\frac{-\sqrt{61}-1}{2}\approx -4,405124838$$

Kvadrat tenglamalaridan foydalanish qoidalari

2 daraja ko‘rsatkichini \sqrt{x+16} ga hisoblang va x+16 ni qiymatni oling.

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

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

$$x+16=x^{2}+2x+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$x+16-x^{2}=2x+1$$

Ikkala tarafdan 2x ni ayirish.

$$x+16-x^{2}-2x=1$$

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

$$-x+16-x^{2}=1$$

Ikkala tarafdan 1 ni ayirish.

$$-x+16-x^{2}-1=0$$

15 olish uchun 16 dan 1 ni ayirish.

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

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -1 ni b va 15 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\times 15}}{2\left(-1\right)}$$

4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+60}}{2\left(-1\right)}$$

1 ni 60 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{61}}{2\left(-1\right)}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{61}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{61}}{-2}$$

x=\frac{1±\sqrt{61}}{-2} tenglamasini yeching, bunda ± musbat. 1 ni \sqrt{61} ga qo'shish.

$$x=\frac{\sqrt{61}+1}{-2}$$

1+\sqrt{61} ni -2 ga bo'lish.

$$x=\frac{-\sqrt{61}-1}{2}$$

x=\frac{1±\sqrt{61}}{-2} tenglamasini yeching, bunda ± manfiy. 1 dan \sqrt{61} ni ayirish.

$$x=\frac{1-\sqrt{61}}{-2}$$

1-\sqrt{61} ni -2 ga bo'lish.

$$x=\frac{\sqrt{61}-1}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{61}-1}{2} x=\frac{\sqrt{61}-1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

2 daraja ko‘rsatkichini \sqrt{x+16} ga hisoblang va x+16 ni qiymatni oling.

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

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

$$x+16=x^{2}+2x+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$x+16-x^{2}=2x+1$$

Ikkala tarafdan 2x ni ayirish.

$$x+16-x^{2}-2x=1$$

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

$$-x+16-x^{2}=1$$

Ikkala tarafdan 16 ni ayirish.

$$-x-x^{2}=1-16$$

-15 olish uchun 1 dan 16 ni ayirish.

$$-x-x^{2}=-15$$

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

$$-x^{2}-x=-15$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-x}{-1}=-\frac{15}{-1}$$

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

$$x^{2}+\left(-\frac{1}{-1}\right)x=-\frac{15}{-1}$$

-1 ni -1 ga bo'lish.

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

-15 ni -1 ga bo'lish.

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

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=15+\left(\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$x^{2}+x+\frac{1}{4}=15+\frac{1}{4}$$

15 ni \frac{1}{4} ga qo'shish.

$$x^{2}+x+\frac{1}{4}=\frac{61}{4}$$

x^{2}+x+\frac{1}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{61}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{61}{4}}$$

Qisqartirish.

$$x+\frac{1}{2}=\frac{\sqrt{61}}{2} x+\frac{1}{2}=-\frac{\sqrt{61}}{2}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=\frac{\sqrt{61}-1}{2} x=\frac{-\sqrt{61}-1}{2}$$