4w+16w^2=80 eching

$$4 w + 16 w ^ { 2 } = 80$$

w uchun yechish

$$w = \frac{\sqrt{321} - 1}{8} \approx 2,114559108$$

$$w=\frac{-\sqrt{321}-1}{8}\approx -2,364559108$$

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.

$$16w^{2}+4w=80$$

Tenglamaning ikkala tarafidan 80 ni ayirish.

$$16w^{2}+4w-80=80-80$$

O‘zidan 80 ayirilsa 0 qoladi.

$$16w^{2}+4w-80=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 16 ni a, 4 ni b va -80 ni c bilan almashtiring.

$$w=\frac{-4±\sqrt{4^{2}-4\times 16\left(-80\right)}}{2\times 16}$$

4 kvadratini chiqarish.

$$w=\frac{-4±\sqrt{16-4\times 16\left(-80\right)}}{2\times 16}$$

-4 ni 16 marotabaga ko'paytirish.

$$w=\frac{-4±\sqrt{16-64\left(-80\right)}}{2\times 16}$$

-64 ni -80 marotabaga ko'paytirish.

$$w=\frac{-4±\sqrt{16+5120}}{2\times 16}$$

16 ni 5120 ga qo'shish.

$$w=\frac{-4±\sqrt{5136}}{2\times 16}$$

5136 ning kvadrat ildizini chiqarish.

$$w=\frac{-4±4\sqrt{321}}{2\times 16}$$

2 ni 16 marotabaga ko'paytirish.

$$w=\frac{-4±4\sqrt{321}}{32}$$

w=\frac{-4±4\sqrt{321}}{32} tenglamasini yeching, bunda ± musbat. -4 ni 4\sqrt{321} ga qo'shish.

$$w=\frac{4\sqrt{321}-4}{32}$$

-4+4\sqrt{321} ni 32 ga bo'lish.

$$w=\frac{\sqrt{321}-1}{8}$$

w=\frac{-4±4\sqrt{321}}{32} tenglamasini yeching, bunda ± manfiy. -4 dan 4\sqrt{321} ni ayirish.

$$w=\frac{-4\sqrt{321}-4}{32}$$

-4-4\sqrt{321} ni 32 ga bo'lish.

$$w=\frac{-\sqrt{321}-1}{8}$$

Tenglama yechildi.

$$w=\frac{\sqrt{321}-1}{8} w=\frac{-\sqrt{321}-1}{8}$$

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.

$$16w^{2}+4w=80$$

Ikki tarafini 16 ga bo‘ling.

$$\frac{16w^{2}+4w}{16}=\frac{80}{16}$$

16 ga bo'lish 16 ga ko'paytirishni bekor qiladi.

$$w^{2}+\frac{4}{16}w=\frac{80}{16}$$

\frac{4}{16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$w^{2}+\frac{1}{4}w=\frac{80}{16}$$

80 ni 16 ga bo'lish.

$$w^{2}+\frac{1}{4}w=5$$

\frac{1}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{8} olish uchun. Keyin, \frac{1}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$w^{2}+\frac{1}{4}w+\left(\frac{1}{8}\right)^{2}=5+\left(\frac{1}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{8} kvadratini chiqarish.

$$w^{2}+\frac{1}{4}w+\frac{1}{64}=5+\frac{1}{64}$$

5 ni \frac{1}{64} ga qo'shish.

$$w^{2}+\frac{1}{4}w+\frac{1}{64}=\frac{321}{64}$$

w^{2}+\frac{1}{4}w+\frac{1}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(w+\frac{1}{8}\right)^{2}=\frac{321}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(w+\frac{1}{8}\right)^{2}}=\sqrt{\frac{321}{64}}$$

Qisqartirish.

$$w+\frac{1}{8}=\frac{\sqrt{321}}{8} w+\frac{1}{8}=-\frac{\sqrt{321}}{8}$$

Tenglamaning ikkala tarafidan \frac{1}{8} ni ayirish.

$$w=\frac{\sqrt{321}-1}{8} w=\frac{-\sqrt{321}-1}{8}$$