-4b^2+22b-4=0 eching

$$- 4 b ^ { 2 } + 22 b - 4 = 0$$

b uchun yechish

$$b = \frac{\sqrt{105} + 11}{4} \approx 5,311737691$$

$$b=\frac{11-\sqrt{105}}{4}\approx 0,188262309$$

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.

$$-4b^{2}+22b-4=0$$

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

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

22 kvadratini chiqarish.

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

-4 ni -4 marotabaga ko'paytirish.

$$b=\frac{-22±\sqrt{484+16\left(-4\right)}}{2\left(-4\right)}$$

16 ni -4 marotabaga ko'paytirish.

$$b=\frac{-22±\sqrt{484-64}}{2\left(-4\right)}$$

484 ni -64 ga qo'shish.

$$b=\frac{-22±\sqrt{420}}{2\left(-4\right)}$$

420 ning kvadrat ildizini chiqarish.

$$b=\frac{-22±2\sqrt{105}}{2\left(-4\right)}$$

2 ni -4 marotabaga ko'paytirish.

$$b=\frac{-22±2\sqrt{105}}{-8}$$

b=\frac{-22±2\sqrt{105}}{-8} tenglamasini yeching, bunda ± musbat. -22 ni 2\sqrt{105} ga qo'shish.

$$b=\frac{2\sqrt{105}-22}{-8}$$

-22+2\sqrt{105} ni -8 ga bo'lish.

$$b=\frac{11-\sqrt{105}}{4}$$

b=\frac{-22±2\sqrt{105}}{-8} tenglamasini yeching, bunda ± manfiy. -22 dan 2\sqrt{105} ni ayirish.

$$b=\frac{-2\sqrt{105}-22}{-8}$$

-22-2\sqrt{105} ni -8 ga bo'lish.

$$b=\frac{\sqrt{105}+11}{4}$$

Tenglama yechildi.

$$b=\frac{11-\sqrt{105}}{4} b=\frac{\sqrt{105}+11}{4}$$

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.

$$-4b^{2}+22b-4=0$$

4 ni tenglamaning ikkala tarafiga qo'shish.

$$-4b^{2}+22b-4-\left(-4\right)=-\left(-4\right)$$

O‘zidan -4 ayirilsa 0 qoladi.

$$-4b^{2}+22b=-\left(-4\right)$$

0 dan -4 ni ayirish.

$$-4b^{2}+22b=4$$

Ikki tarafini -4 ga bo‘ling.

$$\frac{-4b^{2}+22b}{-4}=\frac{4}{-4}$$

-4 ga bo'lish -4 ga ko'paytirishni bekor qiladi.

$$b^{2}+\frac{22}{-4}b=\frac{4}{-4}$$

\frac{22}{-4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$b^{2}-\frac{11}{2}b=\frac{4}{-4}$$

4 ni -4 ga bo'lish.

$$b^{2}-\frac{11}{2}b=-1$$

-\frac{11}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{4} olish uchun. Keyin, -\frac{11}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$b^{2}-\frac{11}{2}b+\left(-\frac{11}{4}\right)^{2}=-1+\left(-\frac{11}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{4} kvadratini chiqarish.

$$b^{2}-\frac{11}{2}b+\frac{121}{16}=-1+\frac{121}{16}$$

-1 ni \frac{121}{16} ga qo'shish.

$$b^{2}-\frac{11}{2}b+\frac{121}{16}=\frac{105}{16}$$

b^{2}-\frac{11}{2}b+\frac{121}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(b-\frac{11}{4}\right)^{2}=\frac{105}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(b-\frac{11}{4}\right)^{2}}=\sqrt{\frac{105}{16}}$$

Qisqartirish.

$$b-\frac{11}{4}=\frac{\sqrt{105}}{4} b-\frac{11}{4}=-\frac{\sqrt{105}}{4}$$

\frac{11}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$b=\frac{\sqrt{105}+11}{4} b=\frac{11-\sqrt{105}}{4}$$