r^2-22r-7=0 eching

$$r ^ { 2 } - 22 r - 7 = 0$$

r uchun yechish

$$r=8\sqrt{2}+11\approx 22,313708499$$

$$r=11-8\sqrt{2}\approx -0,313708499$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$r^{2}-22r-7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -22 ni b va -7 ni c bilan almashtiring.

$$r=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-7\right)}}{2}$$

-22 kvadratini chiqarish.

$$r=\frac{-\left(-22\right)±\sqrt{484-4\left(-7\right)}}{2}$$

-4 ni -7 marotabaga ko'paytirish.

$$r=\frac{-\left(-22\right)±\sqrt{484+28}}{2}$$

484 ni 28 ga qo'shish.

$$r=\frac{-\left(-22\right)±\sqrt{512}}{2}$$

512 ning kvadrat ildizini chiqarish.

$$r=\frac{-\left(-22\right)±16\sqrt{2}}{2}$$

-22 ning teskarisi 22 ga teng.

$$r=\frac{22±16\sqrt{2}}{2}$$

r=\frac{22±16\sqrt{2}}{2} tenglamasini yeching, bunda ± musbat. 22 ni 16\sqrt{2} ga qo'shish.

$$r=\frac{16\sqrt{2}+22}{2}$$

22+16\sqrt{2} ni 2 ga bo'lish.

$$r=8\sqrt{2}+11$$

r=\frac{22±16\sqrt{2}}{2} tenglamasini yeching, bunda ± manfiy. 22 dan 16\sqrt{2} ni ayirish.

$$r=\frac{22-16\sqrt{2}}{2}$$

22-16\sqrt{2} ni 2 ga bo'lish.

$$r=11-8\sqrt{2}$$

Tenglama yechildi.

$$r=8\sqrt{2}+11 r=11-8\sqrt{2}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$r^{2}-22r-7=0$$

7 ni tenglamaning ikkala tarafiga qo'shish.

$$r^{2}-22r-7-\left(-7\right)=-\left(-7\right)$$

O‘zidan -7 ayirilsa 0 qoladi.

$$r^{2}-22r=-\left(-7\right)$$

0 dan -7 ni ayirish.

$$r^{2}-22r=7$$

-22 ni bo‘lish, x shartining koeffitsienti, 2 ga -11 olish uchun. Keyin, -11 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$r^{2}-22r+\left(-11\right)^{2}=7+\left(-11\right)^{2}$$

-11 kvadratini chiqarish.

$$r^{2}-22r+121=7+121$$

7 ni 121 ga qo'shish.

$$r^{2}-22r+121=128$$

r^{2}-22r+121 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(r-11\right)^{2}=128$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(r-11\right)^{2}}=\sqrt{128}$$

Qisqartirish.

$$r-11=8\sqrt{2} r-11=-8\sqrt{2}$$

11 ni tenglamaning ikkala tarafiga qo'shish.

$$r=8\sqrt{2}+11 r=11-8\sqrt{2}$$